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Transient Wave ImagingLili Guadarrama

To cite this version:Lili Guadarrama. Transient Wave Imaging. Analysis of PDEs [math.AP]. Ecole Polytechnique X,2010. English. pastel-00543301

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Ecole Doctorale Polytechnique

These de doctorat

Discipline : Mathematiques Appliquees

presentee par

Lilı Guadarrama Bustos

Imagerie en regime temporel

dirigee par Habib Ammari

Soutenue le 4 juin 2010 devant le jury compose de :

M. Vilmos KOMORNIK Universite de Strasbourg president

M. Yves CAPDEBOSCQ University of Oxford rapporteur

M. Knut SØLNA University of California at Irvine rapporteur

M. Josselin GARNIER Universit Paris VII examinateur

M. Roman NOVIKOV Ecole Polytechnique examinateur

M. Habib AMMARI Ecole Normale Superieure directeur

Transient Wave Imaging

Lili Guadarrama Bustos

Centre de Mathematiques Appliquees, Ecole Polytechnique, 91128

Palaiseau, France

E-mail address: [email protected]

Contents

Introduction 1

Introduction en Francais 3

Acknowledgments 5

Chapter 1. Transient acoustic imaging 71.1. Introduction 71.2. Asymptotic expansions for the Helmholtz equation 81.3. Far- and near-field asymptotic formulas for the transient wave

equation 151.4. Reconstruction methods 171.5. Numerical illustrations 211.6. Concluding remarks 23

Chapter 2. Transient elasticity imaging and time reversal 272.1. Introduction 272.2. Asymptotic expansions 272.3. Far- and near-field asymptotic formulas in the transient regime 392.4. Asymptotic imaging 412.5. Concluding remarks 42

Chapter 3. Transient imaging with limited-view data 433.1. Introduction 433.2. Geometric control 443.3. Imaging algorithms 453.4. Applications to emerging biomedical imaging 473.5. Numerical illustrations 483.6. Concluding remarks 56

Chapter 4. Imaging in visco-elastic media obeying a frequency power-law 654.1. Introduction 654.2. General visco-elastic wave equation 654.3. Green’s function 664.4. Imaging procedure 694.5. Numerical illustrations 714.6. Concluding remarks 71Appendix A: Proof of the approximation formula 72

Bibliography 75

Index 81

v

Introduction

Extensive work has been carried out in the past decade to image the elasticproperties of human soft tissues by inducing motion. This broad field, called elas-ticity imaging or elastography, is based on the initial idea that shear elasticity canbe correlated with the pathology of tissues [60].

There are several techniques that can be classified according to the type ofmechanical excitation chosen (static compression, monochromatic, or transient vi-bration) and the way these excitations are generated (externally or internally).Different imaging modalities can be used to estimate the resulting tissue displace-ments.

A very interesting approach to assessing elasticity is to use the acoustic radi-ation force of an ultrasonic focused beam to remotely generate mechanical vibra-tions in organs. The acoustic force is generated by the momentum transfer fromthe acoustic wave to the medium. The radiation force essentially acts as a dipolarsource. A spatio-temporal sequence of the propagation of the induced transientwave can be acquired, leading to a quantitative estimation of the viscoelastic pa-rameters of the studied medium in a source-free region [33, 34].

Our aim in this thesis is to provide a solid mathematical foundation for thistransient technique and to design accurate methods for anomaly detection usingtransient measurements. We consider both the acoustic and elastic cases. Wedevelop efficient reconstruction techniques from not only complete measurementsbut also from limited-view transient data and adapt them in the case of viscousmedia, where the elastic waves are attenuated and/or dispersed.

We begin with transient imaging in a non-dissipative medium. We developanomaly reconstruction procedures that are based on rigorously established innerand outer time-domain asymptotic expansions of the perturbations in the transientmeasurements that are due to the presence of the anomaly. It is worth mentioningthat in order to approximate the anomaly as a dipole with certain polarizability,one has to truncate the high-frequency component of the far-field measurements.

Using the outer asymptotic expansion, we design a time-reversal imaging tech-nique for locating the anomaly. Based on such expansions, we propose an optimiza-tion problem for recovering geometric properties as well as the physical parametersof the anomaly. We justify both theoretically and numerically that scale separationcan be used to obtain local and precise reconstructions. We show the differencesbetween the acoustic and the elastic cases, namely, the anisotropy of the focal spotand the birth of a near fieldlike effect by time reversing the perturbation due to anelastic anomaly. These interesting findings were experimentally observed and firstreported in [43]. Our asymptotic formalism clearly explains them.

In the case of limited-view transient measurements, we construct Kirchhoff-,back-propagation-, MUSIC-, and arrival time-type algorithms for imaging small

1

2 INTRODUCTION

anomalies. Our approach is based on averaging of the limited-view data, usingweights constructed by the geometrical control method [29]. It is quite robust withrespect to perturbations of the non-accessible part of the boundary. Our mainfinding is that if one can construct accurately the geometric control then one canperform imaging with the same resolution using partial data as using completedata.

We also use our asymptotic formalism to explain how to reconstruct a smallanomaly in a viscoelastic medium from wavefield measurements. The visco-elasticmedium obeys a frequency power-law. For simplicity, we consider the Voigt model,which corresponds to a quadratic frequency loss. By using the stationary phasetheorem, we express the ideal elastic field without any viscous effect in terms ofthe measured field in a viscous medium. We then generalize the imaging tech-niques developed for a purely quasi-incompressible elasticity model to recover theviscoelastic and geometric properties of an anomaly from wavefield measurements.

The thesis is organized as follows. In Chapter 1 we provide a mathematicalfoundation for the acoustic radiation force imaging. From the rigorously estab-lished asymptotic expansions of near- and far-field measurements of the transientwave induced by the anomaly, we design asymptotic imaging methods leading to aquantitative estimation of physical and geometrical parameters of the anomaly.

In Chapter 2 we consider a purely quasi-incompressible elasticity model. Werigorously establish asymptotic expansions of near- and far-field measurements ofthe transient elastic wave induced by a small elastic anomaly. Our proof uses layerpotential techniques for the modified Stokes system. Based on these formulas, wedesign asymptotic imaging methods leading to a quantitative estimation of elasticand geometrical parameters of the anomaly. Using time-reversal, we show how toreconstruct the location and geometric features of the anomaly from the far-fieldmeasurements. We put a particular emphasis on the difference between the acousticand the elastic cases, namely, the anisotropy of the focal spot and the birth of anear fieldlike effect by time reversing the perturbation due to an elastic anomaly.

In Chapter 3 we consider for the wave equation the inverse problem of identify-ing locations of point sources and dipoles from limited-view data. Using as weightsparticular background solutions constructed by the geometrical control method, werecover Kirchhoff-, back-propagation-, MUSIC-, and arrival time-type algorithmsby appropriately averaging limited-view data. We show both analytically and nu-merically that if one can construct accurately the geometric control, then one canperform imaging with the same resolution using limited-view as using full-viewdata.

Chapter 4 is devoted to the problem of reconstructing a small anomaly in aviscoelastic medium from wavefield measurements. Expressing the ideal elasticfield without any viscous effect in terms of the measured field in a viscous medium,we generalize the methods described in Chapter 3 to recover the viscoelastic andgeometric properties of an anomaly from wavefield measurements.

The four chapters of this thesis are self-contained and can be read indepen-dently. Results in this thesis will appear in [4, 8, 11, 37, 61].

Introduction en Francais

L’imagerie d’elasticite, ou elastographie consiste a imager les proprietes visco-elastiques des tissus mous du corps humain en observant la reponse en deformation aune excitation mecanique. Cette problematique a donne lieu dans les dix dernieresannees a de nombreux travaux, motives par la correlation entre presence d’unepathologie et observation d’un contrast d’elasticite [60]. Differentes techniques peu-vent etre mises en oeuvre selon le type d’excitation choisie, et la maniere d’estimerles deformations resultantes.

Parmi les techniques se trouve une tres interessante qui consiste a induire dans letissu mou une onde de deplacement et a observer la propagation de l’onde pendantsa traversee du milieu d’interet. La resolution d’un probleme inverse permet dededuire des donnees de deplacement une estimation de la carte d’elasticite du milieu[33, 34].

L’objectif du travail presente dans ce document est de donner un cadre mathematique rigoureux a ce technique, en meme temps dessiner des methodes effectivespour la detection des anomalies a l’aide des mesures en regime temporel. On aconsidere le cadre acoustic et le cadre elastique. On a developpe des techniquesde reconstruction efficaces pour des mesures completes sur la frontiere mais aussipour des mesures temporelles incompletes, on a adapte ces techniques au cadreviscoelastique, ca veut dire que les ondes sont attenue ou disperse ou le deux.

On commence pour considerer une milieu sans dissipation. On a developpe desmethodes de reconstruction des anomalies qui sont base sur des developpementsasymptotiques de champ proche et de champ lointain, qui sont rigoureusementetablis, du perturbation des mesures cause par l’anomalie. Il faut remarquer quepour approximer l’effet de l’anomalie par un dipole il faut couper les composant dehaut frequence des mesures de champ lointain.

Le developpement asymptotique de champ lointain nous permet de developperune technique de type regression temporel pour localiser l’anomalie. On proposeen utilisant le developpement asymptotique de champ proche une probleme de op-timisation pour recuperer les proprietes geometriques et les parametres physiquesde l’anomalie. On justifie d’une maniere theorique et numerique que la separationdes echelles permet de separer les differentes informations codees aux differentesechelles. On montre les differences entre le cadre acoustique et l’elastique, prin-cipalement la tache focal anisotrope et l’effet de champ proche qu’on obtient enfaisant le retournement temporal de la perturbation cause par l’anomalie. Ces ob-servations ont ete observe experimentalement et reporte pour la premiere fois en[43], les quelles sont bien expliques par nos developpements asymptotiques.

En ce qui concerne le cadre des mesures partiels, on developpe des algorithmesde type Kirchhoff, back-propagation, MUSIC et arrival-time pour localiser l’anomalie.On utilise le methode du control geometrique [29] pour aborder la problematique

3

4 INTRODUCTION EN FRANCAIS

des mesures partiels, comme resultat on obtient une methode qui est robuste en cequi concerne aux perturbations dans la partie de la frontiere qui n’est pas accessi-ble. Si on construit de manier precise le control geometrique, on obtient la memeresolution d’imagerie que dans le cadre des mesures complet.

On utilise les developpements asymptotiques pour expliquer comment recon-struire une petite anomalie dans un milieu visco-elastique a partir des mesures duchamp de deplacement. Dans le milieu visco-elastique la frequence obeit une loide puissance, pour simplicite on considere le modele Voigt qui correspond a unefrequence en puissance deux. On utilise le theoreme de la phase stationnaire pourexprimer le champ dans un milieu sans effet de viscosite, que on nommera champideal , en termes du champ dans un milieu visco-elastique. Apres on generaliseles techniques d’imagerie developpes pour le modele purement elastique quasi in-compressible pour reconstruire les proprietes visco-elastiques et geometriques d’uneanomalie a partir des mesures du champ de deplacement.

Le document s’articule de la facon suivante. Dans le chapitre 1, il est donne uncadre mathematique rigoureux a l’imagerie par force de radiation acoustique. Enutilisant les expressions asymptotiques rigoureusement etablis pour les mesures duchamp proche et lointaine de l’onde temporel cause par l’anomalie, on developpedes methodes asymptotiques d’imagerie qui permet de estimer quantitativement lesparametres physiques et geometriques de l’anomalie.

Dans le chapitre 2 on considere un modele purement elastique quasi incom-pressible. Dans le meme esprit que le chapitre precedent des expansions asympto-tiques sont rigoureusement etablis pour les mesures proche et lointaine de l’ondeelastique en regime temporel induit par une petite anomalie elastique. Dans lesdemonstrations, on utilise des techniques de layer potentiel pour le systeme deStokes modifie. En utilisant les formules on developpe des methodes asymptotiquesd’imagerie qui permet de estimer quantitativement les parametres physiques etgeometriques de l’anomalie. En utilisant une technique de retournement temporelon montre comment reconstruire les proprietes geometriques et localiser l’anomaliea partir des mesures du champ lointaine. On insiste sur les differences entre le cadreacoustique et l’elastique en particulier la tache focal anisotrope et le effet de champproche qu’on obtient en faisant le retournement temporel de la perturbation causepar l’anomalie elastique.

Dans le chapitre 3 on considere pour l’equation d’onde le probleme inverse delocaliser point source et dipoles a partir des mesures partiels. En utilisant dessolutions particuliers construit par le methode de control geometrique comme fonc-tions de poids, on recouvre des algorithmes du type Kirchhoff, back-propagation,MUSIC, arrival-time si on fait une moyen convenable sur les mesures partiels. Onmontre de manier analytique et numerique que si on arrive a construire precisementle control geometrique alors on peut effectuer l’imagerie avec la meme resolution enutilisant mesures partiels ou mesures complet.

Le chapitre 4 est dedie a l’extension des techniques de reconstruction au cadrede la visco-elastique dynamique. A partir d’exprimer le champ ideal en termes desmesures du champ dans une milieu visco-elastique , on generalise les methodes decritdans le Chapitre 3 pour recuperer les proprietes visco-elastiques et geometriques del’anomalie a partir de mesures du champ desplacement.

Les cinq chapitres de cette these sont independants et peuvent etre lus separement.Les resultats de cette these seront publies dans [4, 8, 11, 37, 61].

Acknowledgments

I would like to express my deep and sincere gratitude to Habib, first for accept-ing to be my advisor, and then for his patience and his kindness that he had duringmy thesis. I hope I learned something that will allow me continue in the reseachway. I was very lucky to have such a great person to guide me in my thesis. I wantto thank him for everything he did for me. I want to thank Professor Capdeboscqand Professor Sølna for accepting to be my rapporteurs, for their careful reviewsof this thesis, their remarks and corrections. I want to thank Professor Kolmornik,Garnier and Novikov for accepting to be my referees.

I am also grateful to the Langevin Institut for opening its door to a theoreticalstudy of transient imaging.

Thanks also to all my colleagues who I worked with, Abdul, Sohuir, Vincent,Eli, it was a pleasure to work with all of you.

I want to thank my dear husband, Arturo, I want to thank him for his faithfulsupport.

Finally I want to tanks to all my friends, for all the good moments we shared.Financial support was provided by CONACyT.

5

CHAPTER 1

Transient acoustic imaging

Abstract. This chapter is devoted to provide a solid mathematical founda-

tion for a promising imaging technique based on the acoustic radiation force,

which acts as a dipolar source. From the rigorously established asymptotic ex-pansions of near- and far-field measurements of the transient wave induced by

the anomaly, we design asymptotic imaging methods leading to a quantitative

estimation of physical and geometrical parameters of the anomaly.

1.1. Introduction

An interesting approach to assessing elasticity is to use the acoustic radiationforce of an ultrasonic focused beam to remotely generate mechanical vibrations inorgans [60]. The acoustic force is generated by the momentum transfer from theacoustic wave to the medium. The radiation force essentially acts as a dipolarsource. A spatio-temporal sequence of the propagation of the induced transientwave can be acquired, leading to a quantitative estimation of the viscoelastic pa-rameters of the studied medium in a source-free region [33, 34].

The aim of this chapter is to provide a solid mathematical foundation for thistechnique and to design new methods for anomaly detection using the radiationforce. These reconstruction procedures are based on rigorously established innerand outer asymptotic expansions of the perturbations of the wavefield that are dueto the presence of the anomaly.

To be more precise, suppose that an anomaly D of the form

D = εB + z

is present, where ε is the (small) diameter of D, B is a reference domain, and zindicates the location of D. A spherical wave

Uy(x, t) :=δt=|x−y|4π|x− y|

is generated by a point source located at y far away from z. When this wavehits the anomaly D, it is perturbed. We will derive asymptotic expansions of thisperturbation near and far away from the anomaly as ε tends to 0. In fact, we willderive asymptotic expansions of the perturbation u − Uy after the high frequencycomponent is truncated, where u is the solution to

∂2t u−∇ ·

(χ(R3 \D) + kχ(D)

)∇u = δx=yδt=0 in R

3×]0,+∞[,

u(x, t) = 0 for x ∈ R3 and t 0.

For example, after truncation of the high-frequency component of the solution, thederived asymptotic expansion far away from the anomaly shows that when thespherical wave Uy reaches the anomaly, it is polarized and emits a new wave. The

7

8 1. TRANSIENT ACOUSTIC IMAGING

threshold of the truncation is determined by the diameter of the anomaly and is oforder O(ε−α) for 0 ≤ α < 1.

Derivations of asymptotic expansions in this chapter are rigorous. They arebased on careful and precise estimates of the dependence with respect to the fre-quency of the remainders in associated asymptotic formulas for the Helmholtz equa-tion. Using the outer asymptotic expansion, we design a time-reversal imagingtechnique for locating the anomaly from measurements of the perturbations in thewavefield in the far-field. It turns out that using the far-field measurement we canreconstruct the location and the polarization tensor of the anomaly. However, It isknown that it is impossible to separate geometric features such as the volume fromthe physical parameters using only the polarization tensor. We show that in orderto reconstruct the shape and to separate the physical parameters of the anomalyfrom its volume one should use near-field perturbations of the wavefield. Basedon such expansions, we propose an optimization problem for recovering geometricproperties as well as the parameters of the anomaly. The connection between ourexpansions and reconstruction methods for the wave equation in this chapter andthose for the Helmholtz equation is discussed in some detail.

In connection with this work, we shall mention on one hand the papers [103,15, 62] for the derivations of asymptotic formula for the Helmholtz equation in thepresence of small volume anomalies and on the other hand, the review paper [26]and the recent book [14] on different algorithms in wave imaging.

The chapter is organized as follows. We rigorously derive in Section 1.2 as-ymptotic formulas for the Helmholtz equation and estimate the dependence of theremainders in these formulas with respect to the frequency. Based on these esti-mates, we obtain in Section 1.3 formulas for the transient wave equation that arevalid after truncating the high-frequency components of the fields. These formulasdescribe the effect of the presence of a small anomaly in both the near and farfield. In Section 1.4 we propose different methods for detecting the physical andgeometric parameters of the anomaly. A time-reversal method is proposed to locatethe anomaly and find its polarization tensor from far-field measurements while anoptimization problem is formulated for reconstructing geometric parameters of theanomaly and its conductivity.

1.2. Asymptotic expansions for the Helmholtz equation

In this section we rigorously derive asymptotic formulas for the Helmholtzequation and estimate the dependence of the remainders in these formulas withrespect to the frequency. For doing so, we rely on a layer-potential technique.

1.2.1. Layer potentials. For ω ≥ 0, let

(1.1) Φω(x) = −e√−1ω|x|

4π|x| , x ∈ R3, x 6= 0,

which is the fundamental solution for the Helmholtz operator ∆+ω2. For a boundedLipschitz domain Ω in R

3 and ω ≥ 0, let SωΩ be the single-layer potential for ∆+ω2,that is,

(1.2) SωΩ[ϕ](x) =

∫

∂Ω

Φω(x− y)ϕ(y) dσ(y), x ∈ R3,

1.2. ASYMPTOTIC EXPANSIONS FOR THE HELMHOLTZ EQUATION 9

for ϕ ∈ L2(∂Ω). When ω = 0, S0Ω is the single layer potential for the Laplacian.

Note that u = SωΩ[ϕ] satisfies the Helmholtz equation (∆ + ω2)u = 0 in Ω and in

R3 \ Ω. Moreover, if ω > 0, it satisfies the radiation condition, namely,

(1.3)

∣∣∣∣∂u

∂r−√−1ωu

∣∣∣∣ = O

(r−2

)as r = |x| → +∞ uniformly in

x

|x| .

It is well-known that the normal derivative of the single-layer potential onLipschitz domains obeys the following jump relation

(1.4)∂(SωΩ[ϕ])

∂ν

∣∣∣∣±

(x) =

(± 1

2I + (K−ω

Ω )∗)

[ϕ](x) a.e. x ∈ ∂Ω,

for ϕ ∈ L2(∂Ω), where (K−ωΩ )∗ is the singular integral operator defined by

(K−ωΩ )∗[ϕ](x) = p.v.

∫

∂Ω

∂Φω(x− y)

∂ν(x)ϕ(y)dσ(y).

Here and throughout this chapter the subscripts ± denote the limit from outsideand inside of ∂Ω.

The operator S0Ω is bounded from L2(∂Ω) into H1(∂Ω) and invertible in three

dimensions [102]. Moreover, one can easily see that there exists ω0 > 0 such thatfor ω < ω0

(1.5) ‖SωΩ[ϕ] − S0Ω[ϕ]‖H1(∂Ω) ≤ Cω‖ϕ‖L2(∂Ω)

for all ϕ ∈ L2(∂Ω) where C is independent of ω. It is also well-known that thesingular integral operator (K0

Ω)∗ is bounded on L2(∂Ω) (see [17] for example).Similarly to (1.5), one can see that for there exists ω0 > 0 such that for ω < ω0

‖(K−ωΩ )∗[ϕ] − (K0

Ω)∗[ϕ]‖L2(∂Ω) ≤ Cω‖ϕ‖L2(∂Ω)

for some constant C independent of ω. In view of (1.4), it amounts to

(1.6)

∥∥∥∥∥∂(SωΩ[ϕ])

∂ν

∣∣∣∣±− ∂(S0

Ω[ϕ])

∂ν

∣∣∣∣±

∥∥∥∥∥L2(∂Ω)

≤ Cω‖ϕ‖L2(∂Ω).

1.2.2. Derivations of the asymptotic expansions. Let D be a smoothanomaly with conductivity 0 < k 6= 1 < +∞ inside a background medium withconductivity 1. Suppose that D = εB + z, where B is a domain which plays therole of a reference domain, ε denotes the small diameter of D, and z indicates thelocation of D.

Let y be a point in R3 such that |y − z| >> ε, and let

(1.7) V (x, ω) := Φω(x− y) = −e√−1ω|x−y|

4π|x− y| ,

so that V satisfies

(1.8) ∆V + ω2V = δx=y,

together with the radiation condition (1.3).Let v(x, ω) be the solution to

(1.9) ∇ · (χ(R3 \D) + kχ(D))∇v + ω2v = δx=y

satisfying the radiation condition (1.3). In this section, we derive asymptotic ex-pansion formula for v− V as ε tends to 0. An important feature of the asymptotic


formula derived in this section is a careful estimate of the dependency of the re-mainder term on the frequency.

Put w = v − V . Then w is a unique solution to

(1.10) ∇ · (χ(R3 \D) + kχ(D))∇w + ω2w = (1 − k)∇ · χ(D)∇V in R3

with the radiation condition. In other words, w is the solution to

(1.11)

∆w +ω2

kw = (1 − 1

k)ω2V in D,

∆w + ω2w = 0 in R3 \D,

w|+ − w|− = 0 on ∂D,

∂w

∂ν

∣∣∣∣+

− k∂w

∂ν

∣∣∣∣−

= (k − 1)∂V

∂ν,

w satisfies the radiation condition.

Therefore, w can be represented as

(1.12) w(x, ω) =

(1

k− 1)ω2

∫

D

Φ ω√k(x− y)V (y)dy + S

ω√k

D [ϕ](x), x ∈ D,

SωD[ψ](x), x ∈ R3 \D,

where (ϕ,ψ) ∈ L2(∂D)2 is the solution to the integral equation(1.13)

Sω√k

D [ϕ] − SωD[ψ] = (1 − 1

k)ω2

∫

D

Φ ω√k(· − y)V (y)dy,

k∂S

ω√k

D [ϕ]

∂ν

∣∣∣∣−− ∂SωD[ψ]

∂ν

∣∣∣∣+

= (1 − k)ω2 ∂

∂ν

∫

D

Φ ω√k(· − y)V (y)dy + (1 − k)

∂V

∂ν,

on ∂D. The unique solvability of (1.13) will be shown in the sequel.Let

ϕ(x) = ϕ(εx+ z), x ∈ ∂B,

and define ψ likewise. Then, after changes of variables, (1.13) takes the form

(1.14)

Sεω√

k

B [ϕ] − SεωB [ψ] = F,

k∂S

εω√k

B [ϕ]

∂ν

∣∣∣∣−− ∂SεωB [ψ]

∂ν

∣∣∣∣+

= G,on ∂B,

where

F (x, ω) = (1 − 1

k)εω2

∫

B

Φ εω√k(x− y)V (εy + z)dy,(1.15)

G(x, ω) = (1 − k)εω2 ∂

∂ν

∫

B

Φ εω√k(x− y)V (εy + z)dy + (1 − k)

∂V

∂ν(εx+ z).(1.16)

Define an operator T : L2(∂B) × L2(∂B) → H1(∂B) × L2(∂B) by

(1.17) T (ϕ, ψ) :=

S

εω√k

B [ϕ] − SεωB [ψ], k∂S

εω√k

B [ϕ]

∂ν

∣∣∣∣−− ∂SεωB [ψ]

∂ν

∣∣∣∣+

.


We then decompose T as

(1.18) T = T0 + Tε,

where

(1.19) T0(ϕ, ψ) :=

(S0B [ϕ] − S0

B [ψ], k∂S0

B [ϕ]

∂ν

∣∣∣∣−− ∂S0

B [ψ]

∂ν

∣∣∣∣+

),

and Tε := T − T0. In view of (1.5) and (1.6), we have

(1.20) ‖Tε(ϕ, ψ)‖H1(∂B)×L2(∂B) ≤ Cεω(‖ϕ‖L2(∂B) + ‖ψ‖L2(∂B))

for some constant C independent of ε and ω.Since S0

B : L2(∂B) → H1(∂B) is invertible, we readily see that T0 : L2(∂B) ×L2(∂B) → H1(∂B) × L2(∂B) is invertible. In fact, we have the following lemma.

Lemma 1.1. For (f, g) ∈ H1(∂B) × L2(∂B) let (ϕ, ψ) = T−10 (f, g). Then

ϕ = ψ + (S0B)−1[f ],

ψ =

(k + 1

2(k − 1)I − (K0

B)∗)−1 [

k

k − 1(−1

2I + (K0

B)∗)(S0B)−1[f ] − 1

k − 1g

].

Thanks to (1.18) and (1.20), there is ε0 > 0 such that T is invertible if εω ≤ ε0and

(1.21) T−1 = T−10 + E,

where the operator E satisfies

‖E(f, g)‖L2(∂B)×L2(∂B) ≤ Cεω(‖f‖H1(∂B) + ‖g‖L2(∂B))

for some constant C independent of ε and ω.

Suppose that εω ≤ ε0 < 1. Let (ϕω, ψω) be the solution to (1.14). Then by(1.21) we have

(1.22) (ϕω, ψω) = T−10 (F,G) + E(F,G).

Observe that

(1.23) ‖F‖H1(∂B) ≤ Cεω2.

On the other hand, G can be written as

G(x) = (1 − k)∇V (z, ω) · ν(x) +G1(x),

where G1 satisfies

‖G1‖L2(∂B) ≤ Cεω2.

Therefore, we have

(1.24) (ϕω, ψω) = T−10 (0, (1 − k)∇V (z) · ν) + T−1

0 (F,G1) + E(F,G).

Note that

‖T−10 (F,G1) + E(F,G)‖L2(∂B)×L2(∂B) ≤ Cεω2.


We also need asymptotic expansion of ∂ eϕω

∂ω and ∂ eψω

∂ω . By differentiating bothsides of (1.14) with respect to ω, we obtain(1.25)

Sεω√

k

B

[∂ϕω∂ω

]− SεωB

[∂ψω∂ω

]=∂F

∂ω

− ε

4π√k

∫

∂B

e√−1 εω√

k|·−y|

ϕω(y)dσ(y) +ε

4π

∫

∂B

e√−1εω|·−y|ψω(y)dσ(y),

k∂

∂νS

εω√k

B

[∂ϕω∂ω

] ∣∣∣∣−− ∂

∂νSεωB

[∂ψω∂ω

] ∣∣∣∣+

=∂G

∂ω

− ε

4π√k

∂

∂ν

∫

∂B

e√−1 εω√

k|·−y|

ϕω(y)dσ(y) +ε

4π

∂

∂ν

∫

∂B

e√−1εω|·−y|ψω(y)dσ(y)

on ∂B. One can see from (1.15) and (1.16) that

∂F

∂ω= O(εω) and

∂G1

∂ω= O(εω).

Using the same argument as before, we then obtain

(1.26) (∂ϕω∂ω

,∂ψω∂ω

) = T−10

(0, (1 − k)∇∂V

∂ω(z, ω) · ν

)+O(εω),

where the equality holds in L2(∂B) × L2(∂B).We obtain the following proposition from Lemma 1.1 (with f = 0), (1.24), and

(1.26).

Proposition 1.2. Let (ϕω, ψω) be the solution to (1.14). There exists ε0 > 0such that if εω < ε0, then the following asymptotic expansions hold:

ϕω =

(k + 1

2(k − 1)I − (K0

B)∗)−1

[ν] · ∇V (z, ω) +O(εω2),(1.27)

ψω =

(k + 1

2(k − 1)I − (K0

B)∗)−1

[ν] · ∇V (z, ω) +O(εω2),(1.28)

and

∂ϕω∂ω

=

(k + 1

2(k − 1)I − (K0

B)∗)−1

[ν] · ∇∂V

∂ω(z, ω) +O(εω),(1.29)

∂ψω∂ω

=

(k + 1

2(k − 1)I − (K0

B)∗)−1

[ν] · ∇∂V

∂ω(z, ω) +O(εω),(1.30)

where all the equalities hold in L2(∂B).

We are now ready to derive the inner expansion of w = v − V . Let Ω be a set

containing D and let Ω = 1εΩ− z. After changes of variables, (1.12) takes the form

(1.31)

w(εx+z, ω) =

(1

k− 1)ε2ω2

∫

B

Φ εω√k(x− y)V (εy + z)dy + εS

εω√k

B [ϕω](x), x ∈ B,

εSεωB [ψω](x), x ∈ Ω \B.Since ∥∥S

εω√k

B [ϕω] − S0B [ϕω]

∥∥H1(∂B)

≤ Cεω‖ϕω‖L2(∂B),


we have

w(εx+ z, ω) =

εS0B [ϕω](x) +O(ε2ω2), x ∈ B,

εS0B [ψω](x) +O(ε2ω), x ∈ Ω \B.

Here we assumed that ω ≥ 1 since the case when ω < 1 is much easier to handle.It then follows from (1.27) and (1.28) that

w(εx+ z, ω) = εS0B

(k + 1

2(k − 1)I − (K0

B)∗)−1

[ν](x) · ∇V (z, ω) +O(ε2ω2), x ∈ Ω.

On the other hand, we have

∂w

∂ω(εx+ z, ω) =

εSεω√

k

B

[∂ϕω∂ω

](x) +O(ε2ω), x ∈ B,

εSεω√

k

B

[∂ψω∂ω

](x) +O(ε2ω), x ∈ Ω \B.

Therefore, we have from (1.29) and (1.30)

∂w

∂ω(εx+z, ω) = εS0

B

(k + 1

2(k − 1)I − (K0

B)∗)−1

[ν](x)·∇∂V

∂ω(z, ω)+O(ε2ω), x ∈ Ω.

Let

v1(x) := S0B

(k + 1

2(k − 1)I − (K0

B)∗)−1

[ν](x).

Note that v1 is a vector-valued function. It is well-known that v1 is the solution to

(1.32)

∆v1 = 0 in R3 \B,

∆v1 = 0 in B,

v1|− − v1|+ = 0 on ∂B,

k∂v1∂ν

∣∣∣∣−− ∂v1

∂ν

∣∣∣∣+

= (k − 1)ν on ∂B,

v1(x) = O(|x|−2) as |x| → +∞.

We finally obtain the following theorem.

Theorem 1.3. Let Ω be a bounded domain containing D and let

(1.33) R(x, ω) = v(x, ω) − V (x, ω) − εv1

(x− z

ε

)· ∇V (z, ω).

There exists ε0 > 0 such that if εω < ε0, then

(1.34) R(x, ω) = O(ε2ω2), ∇xR(x, ω) = O(εω2) x ∈ Ω.

Moreover,

(1.35)∂R

∂ω(x, ω) = O(ε2ω), ∇x

(∂R

∂ω

)(x, ω) = O(εω) x ∈ Ω.

Note that the estimates for ∇xR in (1.34) and ∇x(∂R∂ω ) in (1.35) can be derived

using (1.31).


Based on Theorem 1.3 we can easily derive an asymptotic expansion of v(x, ω)−V (x, ω) for |x−z| ≥ C > 0 for some constant C. For doing so, we first define the po-larization tensor M = M(k,B) associated with the domain B and the conductivitycontrast k, 0 < k 6= 1 < +∞, as follows (see [17]):

(1.36) M(k,B) := (k − 1)

∫

B

∇(v1(x) + x) dx.

It should be noticed that the polarization tensor M can be explicitly computedfor balls and ellipsoids in three-dimensional space. We also list important propertiesof M [17]:

(i) M is symmetric.(ii) If k > 1, thenM is positive definite, and it is negative definite if 0 < k < 1.(iii) The following Hashin-Shtrikman bounds

(1.37)

1

k − 1trace(M) ≤ (2 +

1

k)|B|,

(k − 1) trace(M−1) ≤ 2 + k

|B| ,

hold [80, 39], where trace denotes the trace of a matrix.

It is worth mentioning that the equality in the second inequality in (1.37) holds ifand only if B is an ellipsoid [72].

Note that u := v − V satisfies

(∆ + ω2)u = (k − 1)∇ · χ(D)∇v,with the radiation condition. Therefore, using the Lipmann-Schwinger integralrepresentation

v(x, ω) − V (x, ω) = (1 − k)

∫

D

∇v(y, ω) · ∇Φω(x− y) dy,

together with the asymptotic expansion of v in D in Theorem 1.3, we obtain thatfor x away from z, there exists ε0 > 0 such that if εω < ε0, then

v(x, ω) − V (x, ω) = (1 − k)

∫

D

(∇V (y, ω) + ∇v1(

y − z

ε) · ∇V (z, ω)

)· ∇Φω(x− y) dy

+ O(ε4ω3).

Now if we approximate ∇V (y, ω) and ∇Φω(x− y) for y ∈ D by ∇V (z, ω) and∇Φω(x− z), respectively, we obtain the following theorem.

Theorem 1.4. Let Ω′ be a compact region away from D (dist(Ω′,D) ≥ C > 0for some constant C) and let

(1.38) R(x, ω) = v(x, ω) − V (x, ω) + ε3∇V (z, ω)M(k,B)∇Φω(x− z).


(1.39) R(x, ω) = O(ε4ω3), x ∈ Ω′.

Moreover,

(1.40)∂R

∂ω(x, ω) = O(ε4ω2), x ∈ Ω′.

1.3. FAR- AND NEAR-FIELD ASYMPTOTICS 15

Note that, in view of the asymptotic formulae derived in [63] for the case ofa circular anomaly, the range of frequencies for which formula (1.39) is valid isoptimal.

1.3. Far- and near-field asymptotic formulas for the transient waveequation

Let D be a smooth anomaly with conductivity 0 < k 6= 1 < +∞ inside abackground medium with conductivity 1. Suppose that D = εB + z as before.

Let y be a point in R3 such that |y − z| ≥ C > 0 for some constant C. Define

(1.41) Uy(x, t) :=δt=|x−y|4π|x− y| ,

where δ is the Dirac mass.Uy is the Green function associated with the retarded layer potentials and

satisfies [53, 58]

(∂2t − ∆)Uy(x, t) = δx=yδt=0 in R

3 × R,

Uy(x, t) = 0 for x ∈ R3 and t 0.

For ρ > 0, we define the operator Pρ on tempered distributions by

(1.42) Pρ[ψ](t) =

∫

|ω|≤ρe−

√−1ωtψ(ω) dω,

where ψ denotes the Fourier transform of ψ. The operator Pρ truncates the high-frequency component of ψ. Since

Uy(x, ω) = V (x, ω) :=e√−1ω|x−y|

4π|x− y|using the notation in (1.7), we have

(1.43) Pρ[Uy](x, t) =

∫

|ω|≤ρe−

√−1ωtV (x, ω)dω =

ψρ(t− |x− y|)4π|x− y| for x 6= y,

where

(1.44) ψρ(t) :=2 sin ρt

t=

∫

|ω|≤ρe−

√−1ωtdω.

One can easily show that Pρ[Uy] satisfies

(1.45) (∂2t − ∆)Pρ[Uy](x, t) = δx=yψρ(t) in R

3 × R.

We consider the wave equation in the whole three-dimensional space with ap-propriate initial conditions:

(1.46)

∂2t u−∇ ·

(χ(R3 \D) + kχ(D)

)∇u = δx=yδt=0 in R

3×]0,+∞[,

u(x, t) = 0 for x ∈ R3 and t 0.

The purpose of this section is to derive asymptotic expansions for Pρ[u −Uy](x, t). For that purpose, we observe that

(1.47) Pρ[u](x, t) =

∫

|ω|≤ρe−

√−1ωtv(x, ω)dω,


where v is the solution to (1.9). Therefore, according to Theorem 1.3, we have

Pρ[u− Uy](x, t) − εv1

(x− z

ε

)· ∇Pρ[Uy](x, t) =

∫

|ω|≤ρe−

√−1ωtR(x, ω)dω.

Suppose that |t| ≥ c0 for some positive number c0 (c0 is of order the distancebetween y and z). Then, we have by an integration by parts∣∣∣∣∣

∫

|ω|≤ρe−

√−1ωtR(x, ω)dω

∣∣∣∣∣ =∣∣∣∣∣1

t

∫

|ω|≤ρ

d

dωe−


∣∣∣∣∣

≤ 1

|t| (|R(x, ρ)| + |R(x,−ρ)|) +

∫

|ω|≤ρ

∣∣∣∣∂

∂ωR(x, ω)

∣∣∣∣ dω

≤ Cε2ρ2.

Since εv1(x−zε

)· ∇Pρ[Uy] = O(ερ), we arrive at the following theorem.

Theorem 1.5. Suppose that ρ = O(ε−α) for some α < 1. Then

Pρ[u− Uy](x, t) = εv1

(x− z

ε

)· ∇Pρ[Uy](x, t) +O(ε2(1−α)).

We now derive a far-field asymptotic expansion for Pρ[u− Uy]. Define

(1.48) Uz(x, t) :=δt=|x−z|4π|x− z| .

We have

Pρ[Uz](x, t) =

∫

|ω|≤ρe−

√−1ωtΦω(x− z) dω.

From Theorem 1.4, we compute∫

|ω|≤ρe−

√−1ωt(v(x, ω) − V (x, ω)) dω

= −ε3∫

|ω|≤ρe−

√−1ωt∇V (z, ω)M(k,B)∇Φω(x− z) dω

+

∫

|ω|≤ρe−

√−1ωtR(x, ω) dω,

where the remainder is estimated by∫

|ω|≤ρe−

√−1ωtR(x, ω) dω = O(ε4(1−

34α)).

Since∫

|ω|≤ρe−

√−1ωt∇V (z, ω)M(k,B)∇Φω(x− z) dω

=

∫

R

∇(ψρ(t− τ − |x− z|)

4π|x− z| )M(k,B)∇(ψρ(τ − |z − y|)

4π|z − y| ) dτ,

and

ε3∫

R

∇Pρ[Uz](x, t− τ) ·M(k,B)∇Pρ[Uy](z, τ) dτ = O(ε3ρ2),

the following theorem holds.

1.4. RECONSTRUCTION METHODS 17

Theorem 1.6. Suppose that ρ = O(ε−α) for some α < 1. Then for |x − z| ≥C > 0, the following far-field expansion holds

Pρ[u−Uy](x, t) = −ε3∫

R

∇Pρ[Uz](x, t−τ) ·M(k,B)∇Pρ[Uy](z, τ) dτ +O(ε4(1−34α))

for x away from z.

It should be noted that Theorem 1.6 says that the perturbation due to theanomaly is (approximately) a wave emitted from the point z at t = T := |z − y|.The anomaly behaves then like a dipolar source. This is the key point of ourapproach for designing time-reversal imaging procedure in the next section. We alsoemphasize that the approximation holds after truncation of the frequencies higherthan ε−α (α < 1). This has an important meaning in relation to the resolutionlimit in imaging as explained in the next section. Moreover, from the optimality ofthe range of frequencies for which formula (1.38) is valid, it follows that α < 1 isindeed the optimal exponent.

1.4. Reconstruction methods

A model problem for the acoustic radiation force imaging is (1.46), where yis the location of the pushing ultrasonic beam. The transient wave u(x, t) is theinduced wave. The inverse problem is to reconstruct the shape and the conductivityof the small anomaly D from either far-field or near-field measurements of u.

1.4.1. Time-reversal. Let w(x, t) := u(x, t)−Uy(x, t). We present a methodfor detecting the location z of the anomaly from measurements of w for x away fromz. To detect the anomaly one can use a time-reversal technique. The main idea oftime-reversal is to take advantage of the reversibility of the wave equation in a non-dissipative unknown medium in order to back-propagate signals to the sources thatemitted them. See [54, 41, 89, 55]. Some interesting mathematical works startedto investigate different aspects of time-reversal phenomena: see, for instance, [28]for time-reversal in the time-domain, [51, 84, 64, 44, 45] for time-reversal in thefrequency domain, and [57, 36] for time-reversal in random media.

In the context of anomaly detection, one measures the perturbation of thewave on a closed surface surrounding the anomaly, and retransmits it through thebackground medium in a time-reversed chronology. Then the perturbation willtravel back to the location of the anomaly.

Suppose that we are able to measure the perturbation w and its normal deriv-ative at any point x on a sphere S englobing the anomaly D. The time-reversaloperation is described by the transform t 7→ t0− t. Both the perturbation w and itsnormal derivative on S are time-reversed and emitted from S. Then a time-reversedperturbation, denoted by wtr, propagates inside the volume Ω surrounded by S.Taking into account the definition (1.48) of the outgoing fundamental solution, spa-tial reciprocity and time reversal invariance of the wave equation, the time-reversedperturbation wtr due to the anomaly D in Ω should be defined as follows.

Definition 1.7. The time-reversed perturbation is given by

wtr(x, t) =

∫

R

∫

S

[Ux(x

′, t−s)∂w∂ν

(x′, t0−s)−∂Ux∂ν

(x′, t−s)w(x′, t0−s)]dσ(x′) ds,


where

Ux(x′, t− s) =

δt=s+|x−x′|4π|x− x′| .

However, with the high frequency component of w truncated, we take the fol-lowing definition:

(1.49)

wtr(x, t) =

∫

R

∫

S

[Ux(x

′, t− s)∂Pρ[u− Uy]

∂ν(x′, t0 − s)

−∂Ux∂ν

(x′, t− s)Pρ[u− Uy](x′, t0 − s)

]dσ(x′) ds .

According to Theorem 1.6, we have

Pρ[u− Uy](x, t) ≈ −ε3∫

R

∇Pρ[Uz](x, t− τ) · p(z, τ) dτ

where

(1.50) p(z, τ) = M(k,B)∇Pρ[Uy](z, τ).Therefore it follows that

wtr(x, t) ≈ −ε3∫

R

p(z, τ) ·∫

R

∫

S

[Ux(x

′, t− s)∂∇zPρ[Uz]

∂ν(x′, t0 − s− τ)

−∂Ux∂ν

(x′, t− s)∇zPρ[Uz](x′, t0 − s− τ)

]dσ(x′) ds dτ,

≈ −ε3∫

R

p(z, τ) · ∇z

∫

R

∫

S

[Ux(x

′, t− s)∂Pρ[Uz]

∂ν(x′, t0 − s− τ)

−∂Ux∂ν

(x′, t− s)Pρ[Uz](x′, t0 − s− τ)

]dσ(x′) ds dτ.

Multiplying the equation(∂2s − ∆x′

)Ux(x

′, t− s) = δs=tδx′=x

by Pρ[Uz](x′, t0 − τ − s), integrating by parts, and using the equation

(∂2s − ∆x′

)Pρ[Uz](x

′, t0 − τ − s) = ψρ(s− t0 + τ)δx′=z in R3 × R,

we have

(1.51)

∫

R

∫

S

[Ux(x

′, t− s)∂Pρ[Uz]

∂ν(x′, t0 − s− τ)

−∂Ux∂ν

(x′, t− s)Pρ[Uz](x′, t0 − s− τ)

]dσ(x′) ds

= Pρ[Uz](x, t0 − τ − t) − Pρ[Uz](x, t− t0 + τ).

It then follows that

(1.52) wtr(x, t) ≈ −ε3∫

R

p(z, τ) ·∇z

[Pρ[Uz](x, t0−τ− t)−Pρ[Uz](x, t− t0 +τ)

]dτ.

The formula (1.52) can be interpreted as the superposition of incoming andoutgoing waves, centered on the location z of the anomaly. To see it more clearly,let us assume that p(z, τ) is concentrated at τ = T := |z − y|, which is reasonable

1.4. RECONSTRUCTION METHODS 19

since p(z, τ) = M(k,B)∇Pρ[Uy](z, τ) peaks at τ = T . Under this assumption, theformula (1.52) takes the form

(1.53) wtr(x, t) ≈ −ε3p · ∇z

[Pρ[Uz](x, t0 − T − t) − Pρ[Uz](x, t− t0 + T )

],

where p = p(z, T ). It is clearly sum of incoming and outgoing spherical waves.Formula (1.53) has an important physical interpretation. By changing the

origin of time, T can be set to 0 without loss of generality. By taking Fouriertransform of (1.52) over the time variable t, we obtain that

(1.54) wtr(x, ω) ∝ ε3p · ∇(

sin(ω|x− z|)|x− z|

),

where ω is the wavenumber. This shows that the anti-derivative of time-reversalperturbation wtr focuses on the location z of the anomaly with a focal spot sizelimited to one-half the wavelength which is in agreement with the Rayleigh resolu-tion limit. It should be pointed out that in the frequency domain, (1.54) is validonly for λ = 2π/ω ε, ε being the characteristic size of the anomaly. In fact,according to Theorem 1.6, it is valid for frequencies less than O(ε−α) for α < 1.

In the frequency domain, suppose that one measures the perturbation v − Vand its normal derivative on a sphere S englobing the anomaly D. To detect theanomaly D one computes

w(x, ω) :=

∫

S

[Φω(x− x′)

∂(v − V )

∂ν(x′, ω) − (v − V )(x′, ω)

∂Φω∂ν

(x− x′)

]dσ(x′),

in the domain Ω surrounded by S. Observe that w(x, ω) is a solution to theHelmholtz equation: (∆ + ω2)w = 0 in Ω.

An identity parallel to (1.51) can be derived in the frequency domain. Indeed,it plays a key role in achieving the resolution limit. Applying Green’s theorem toΦω(x− x′) and Φω(z − x′), we have

(1.55)

∫

S

[Φω(x− x′)

∂Φω∂ν

(z − x′) − Φω(z − x′)∂Φω∂ν

(x− x′)

]dσ(x′)

= 2√−1=mΦω(z − x).

In view of (1.55), we immediately find from the asymptotic expansion in Theorem1.4 that

(1.56) (v − V )(x, ω) ∝ ε3p · ∇(

sin(ω|x− z|)|x− z|

),

where p = M(k,B)∇V (z, ω). The above approximation shows that the anti-derivative of w(x, ω) has a peak at the location z of the anomaly and also provesthe Rayleigh resolution limit. Note that (1.54) is in a good agreement with (1.56)even though the high-frequency component has been truncated.

It is also worth noticing that a formula similar to (1.56) can be derived in aninhomogeneous medium Ω surrounded by S. We have

∫

S

[G(x− x′, ω)

∂G∂ν

(x′ − z, ω) − G(x′ − z, ω)∂G∂ν

(x− x′, ω)

]dσ(x′)

= 2√−1=mG(x− z, ω),(1.57)

where G is the Green function in the inhomogeneous medium Ω. Identity (1.57)shows that the sharper the behavior of =mG at z is, the higher is the resolution.


It would be quite interesting to see how the behavior of =mG depends on theheterogeneity of the medium.

Once the location z of the anomaly is found, the polarization tensor associ-ated with the anomaly D can be found using the formula in Theorem 1.6. SinceM(k,D) = ε3M(k,B), we minimize over symmetric positive matrices M(k,D) thequantity

L∑

l=1

∣∣∣∣Pρ[u− Uy](xl, t) +

∫

R

∇Pρ[Uz](xl, t− τ)M(k,D) · ∇Pρ[Uy](z, τ) dτ∣∣∣∣,

for L measurement points x1, . . . , xL. It is worth emphasizing that the polarizationtensor M(k,D) contains the mixed information of volume |D| and the conductivityk of the anomaly and it is not possible to separate these two information from M .

However, from the near-field measurements, the shape and the conductivity ofthe anomaly D can be approximately reconstructed.

1.4.2. Kirchhoff imaging. Suppose that |z − y| 1 and |x− z| 1. Then

(1.58) v(x, ω) − V (x, ω) ≈ −ω2ε3

16π2

(z − y)M(k,B)(z − x)

|z − y|2|z − x|2 e−√−1ωz·( y

|y|+x|x| ),

which holds for a broadband of frequencies. Then, for a given search point zS , theKirchhoff imaging functional can be written as

IKI(zS ,

x

|x| ) :=1

L

∑

ωl,l=1,...,L

1

ω2l

e√−1ωlz

S ·( y|y|+

x|x| )(v(x, ωl) − V (x, ωl)),

where L is the number of frequencies (ωl). See [52] and the references therein.In view of (1.58), we have

IKI(zS ,

x

|x| ) ≈ Cd

∫

ω

e√−1ωl(z

S−z)·( y|y|+

x|x| ) dω,

for some constant Cd independent of ω and zS and therefore,

IKI(zS ,

x

|x| ) ≈ Cdδ(zS−z)·( y|y|+

x|x| )=0.

Hence, to determine the location z of the anomaly, one needs three different mea-surement directions x/|x|.

1.4.3. Back-propagation imaging. From single frequency measurements,one can detect the anomaly using a back-propagation-type algorithm. Let θl =xl/|xl| for l = 1, . . . , L, be L measurement directions. For a given search point zS ,the back-propagation imaging functional is given by

IBP(zS) :=1

L

∑

θl,l=1,...,L

e√−1ωzS ·( y

|y|+θl)(v(rθl, ω) − V (rθl, ω)), r 1.

Since for sufficiently large L, since

1

L

L∑

l=1

e√−1ωθl·x ≈

j0(ω|x|) for d = 3,

J0(ω|x|) for d = 2,

1.5. NUMERICAL ILLUSTRATIONS 21

where j0 is the spherical Bessel function of order zero and J0 is the Bessel functionof the first kind and of order zero, it follows from (1.58) that

IBP(zS) ≈ Cd

j0(ω|z − zS |) for d = 3,

J0(ω|z − zS |) for d = 2.

for some constant Cd independent of zS .Note that IBP uses a single frequency which can be selected as the highest one

among those that maximize the signal-to-noise ratio.

1.4.4. Near-field imaging. In view of Theorem 1.5, to reconstruct the shapeand the conductivity of the anomaly D we solve analogously to [10] the followingminimization problem. Suppose that the location z of the anomaly D = z+εB andits characteristic size ε are known. Let W be a domain containing D and define thefunctional

L(f, k) =1

2∆T

∫ T+∆T2

T−∆T2

∥∥∥∥Pρ[u− Uy](x, t) − εv1

(x− z

ε

)· ∇Pρ[Uy](x, t)

∥∥∥∥2

L2(W )

dt

+ β

∫

W

|∇f(x)| dx,

where k is the conductivity of D, β is a regularization parameter, f is the binaryrepresentation of D, i.e.,

f(x) =

1 if x ∈ D,

−1 if x /∈ D,

and v1 is the function corresponding to B as defined in (1.32). Here it suffices totake ∆T to be of order O( ε√

k). We then minimize over binary functions f and

constants 0 < k < +∞(1.59) min

k,fL(f, k)

subject to (1.32). We may relax the minimization problem (1.59) to function ofbounded variation. We refer to [10] for the details.

Note that we have to choose a window W that is not so small to preserve somestability and not so big so that we can gain some accuracy. We refer to [9] for adiscussion on the critical size of the window W that switches between far-field andnear-field reconstructions.

1.5. Numerical illustrations

To illustrate our main findings in this chapter, we first tested the accuracy ofthe derived asymptotic expansions. Then we implemented the imaging algorithmsfor anomaly detection.

The configuration is the following: a spherical anomaly of radius 0.05 andphysical parameter k = 3 is placed at z = (−0.1, 0, 0). The source is at y = (3, 0, 0).To truncate the high frequencies, we took ρ = 2.15 or equivalently α = 1/3.

Figure 1.1 shows comparisons between the fields computed by the asymptoticformulas and by the direct Freefem++ code. The Freefem++ code is based on afinite element discretization in space and a finite difference scheme in time. Wehave chosen a Crank-Nicolson scheme with step ∆t = 0.01. The near fields werecomputed at x = (−0.3, 0, 0) and the far-fields were computed at x = (−8, 0, 0).


The fields obtained from the asymptotic formulas are in good agreement with thosecomputed by the Freefem++ code.

−5 0 5 10 15−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3x 10

−3

Time

Ma

gn

itu

de

Freefem++Asymptotic formula

0 2 4 6 8 10 12 14 16−8

−6

−4

−2

0

2

4

6

8x 10

−5

Time

Ma

gn

itu

de

Figure 1.1. Comparisons between the near fields (on the left) andbetween the far-fields (on the right).

Now we turn to imaging. Figure 1.2 shows the performance of the time-reversalfor detecting the anomaly.

Figure 1.2. Detection result using the time-reversal technique.Here ’*’ shows the transceiver location.

Consider a linear array of 58 receivers placed parallel to the y-axis and spacedby half a wavelength. Figure 1.3 shows the detection result by back-propagation.

Now, consider receivers located at

[4λ cos(π/4), 4λ sin(π/4), 0], [4λ cos(π/4),−4λ sin(π/4), 0], [4λ cos(π/4), 0, 4λ sin(π/4)].

Figures 1.4, 1.5, and 1.6 show the results of the Kirchhoff imaging functionalsfor these three different receiver locations. The position of the anomaly is obtainedas the intersection of the three planes where each of the Kirchhoff functional attainsits maximum. See Figure 1.8.

1.6. CONCLUDING REMARKS 23

Figure 1.3. Real and imaginary parts of the Back-propagationfunctional. Here ’*’ and ’+’ respectively show the transceiver andreceiver locations.

Figure 1.4. Real and imaginary part of the Kirchhoff functionalwhen the receiver is at [4λ cos(π/4), 4λ sin(π/4), 0], ’*’ indicatesthe transceiver location and ’+’ the receiver location.

Figure 1.5. Real and imaginary part of the Kirchhoff functionalwhen the receiver is at [4λ cos(π/4),−4λ sin(π/4), 0].

1.6. Concluding remarks

In this chapter, based on careful estimates of the dependence with respect to thefrequency of the remainders in asymptotic formulas for the Helmholtz equation, wehave rigorously derived the effect of a small conductivity anomaly on transient wave.We have provided near- and far-field asymptotic expansions of the perturbationin the wavefield after truncating its high-frequency component. The threshold ofthe frequency truncation is of order ε−α (α < 1) where ε is the diameter of the


Figure 1.6. Real and imaginary part of the Kirchhoff functionalwhen the receiver is at the position [4λ cos(π/4), 0, 4λ sin(π/4)].

Figure 1.7. Sum of the Real and the imaginary parts of the Kirch-hoff functional.

Figure 1.8. Intersection of the three planes where the real parts ofthe Kirchhoff functionals attain their maximum for three differentreceivers.


anomaly. We have also designed a time-reversal imaging technique for locatingthe anomaly from far-field measurements of the perturbations in the wavefield andreconstructing its polarization tensor. Using a near-field asymptotic formula, wehave proposed an optimization problem to reconstruct the shape and to separatethe physical parameters of the anomaly from its volume. The connection betweenour expansions and reconstruction methods for the wave equation in this chapterand those for the Helmholtz equation has been discussed.

The method and the results of this chapter will be generalized in Chapter 2 todynamic elastic imaging which has important applications in medical imaging [34]as well as in seismology [1].

CHAPTER 2

Transient elasticity imaging and time reversal

Abstract. In this chapter we consider a purely quasi-incompressible elastic-

ity model. We rigorously establish asymptotic expansions of near- and far-field

measurements of the transient elastic wave induced by a small elastic anom-aly. Our proof uses layer potential techniques for the modified Stokes system.

Based on these formulas, we design asymptotic imaging methods leading to a

quantitative estimation of elastic and geometrical parameters of the anomaly.

2.1. Introduction

In this chapter, we neglect the viscosity effect of tissues and only consider apurely quasi-incompressible elasticity model. We derive asymptotic expansions ofthe perturbations of the elastic wavefield that are due to the presence of a smallanomaly in both the near- and far-field regions as the size of the anomaly goesto zero. Then we design an asymptotic imaging method leading to a quantitativeestimation of the shear modulus and shape of the anomaly from near-field measure-ments. Using time-reversal, we show how to reconstruct the location and geometricfeatures of the anomaly from the far-field measurements. We put a particularemphasis on the difference between the acoustic and the elastic cases, namely, theanisotropy of the focal spot and the birth of a near-field like effect by time reversingthe perturbation due to an elastic anomaly.

The results of this chapter extend those in Chapter 1 to transient wave propa-gation in elastic media.

The chapter is organized as follows. In Section 2.2 we rigorously derive asymp-totic formulas for quasi-incompressible elasticity and estimate the dependence ofthe remainders in these formulas with respect to the frequency. Based on theseestimates, we obtain in Section 2.3 formulas for the transient wave equation thatare valid after truncating the high-frequency components of the elastic fields. Theseformulas describe the effect of the presence of a small elastic anomaly in both thenear- and far-field. We then investigate in Section 2.4 the use of time-reversal forlocating the anomaly and detecting its overall geometric and material parametersvia the viscous moment tensor. An optimization problem is also formulated forreconstructing geometric parameters of the anomaly and its shear modulus fromnear-field measurements.

2.2. Asymptotic expansions

We suppose that an elastic medium occupies the whole space R3. Let the

constants λ and µ denote the Lame coefficients of the medium, that are the elasticparameters in absence of any anomaly. With these constants, Lλ,µ denotes the

27

28 2. TRANSIENT ELASTICITY IMAGING

linear elasticity system, namely

(2.1) Lλ,µu := µ∆u + (λ+ µ)∇∇ · u.The traction on a smooth boundary ∂Ω is given by the conormal derivative ∂u/∂νassociated with Lλ,µ,

(2.2)∂u

∂ν:= λ(∇ · u)N + µ∇uN,

where N denotes the outward unit normal to ∂Ω. Here ∇ denotes the symmetricgradient, i.e.,

(2.3) ∇u := ∇u + ∇uT ,

where the superscript T denotes the transpose.The time-dependent linear elasticity system is given by

(2.4) ∂2t u − Lλ,µu = 0.

The fundamental solution or the Green function for the system (2.4) is given byG = (Gij) where(2.5)

Gij =1

4π

3γiγj − δijr3

H√µ√λ+2µ

(x, t)+1

4π(λ+ 2µ)

γiγjrδt= r√

λ+2µ− 1

4πµ

γiγj − δijr

δt= r√µ.

Here r = |x|, γi = xi/r, δij denotes the Kronecker symbol, δ denotes the Dirac

delta function, and H√µ√λ+2µ

(x, t) is defined by

(2.6) H√µ√λ+2µ

(x, t) :=

t ifr√

λ+ 2µ< t <

r√µ,

0 otherwise.

Note that (1/r3)H√µ√λ+2µ

(x, t) behaves like 1/r2 for times (r/√λ+ 2µ) < t <

(r/√µ). See [1].

Suppose that there is an elastic anomaly D, given by D = εB + z, which has

the elastic parameters (λ, µ). Here B is a C2-bounded domain containing the origin,z the location of the anomaly, and ε a small positive parameter representing theorder of magnitude of the anomaly size.

For a given point source y away from the anomaly D and a constant vectora, we consider the following transient elastic wave problem in the presence of ananomaly:

(2.7)

∂2t u − Lλ,µu = δt=0δx=ya in (R3 \D) × R,

∂2t u − Lλ,µu = 0 in D × R,

u∣∣+− u

∣∣− = 0 on ∂D × R,

∂u

∂ν

∣∣+− ∂u

∂ν

∣∣− = 0 on ∂D × R,

u(x, t) = 0 for x ∈ R3 and t 0,

where ∂u/∂ν and ∂u/∂ν denote the conormal derivatives on ∂D associated re-spectively with Lλ,µ and Lλ,µ. Here and throughout this chapter the subscripts ±denote the limit from outside and inside D, respectively.

2.2. ASYMPTOTIC EXPANSIONS 29

As was observed in [60, 82], the Poisson ratio of human tissues is very close

to 1/2, which amounts to λ/µ and λ/µ being very large. So we seek for a good

approximation of the problem (2.7) as λ and λ go to +∞. To this end, let

p :=

λ∇ · u in (R3 \ D) × R,

λ∇ · u in D × R.

One can show by modifying a little the argument in [10] that as λ and λ go to +∞with λ/λ of order one, the displacement field u can be represented in the form ofthe following series:

u(x, t) = u0(x, t) + (1

λχ(R3 \D) +

1

λχ(D)) u1(x, t)

+ (1

λ2χ(R3 \D) +

1

λ2χ(D)) u2(x, t) + . . . ,

p = p0 + (1

λχ(R3 \D) +

1

λχ(D)) p1 + (

1

λ2χ(R3 \D) +

1

λ2χ(D)) p2 + . . . ,

where the leading-order term (u0(x, t), p0(x)) is solution to the following homoge-neous time-dependent Stokes system

(2.8)

∂2t u0 −∇ · (µχ(D) + µχ(R3 \D))∇u0 −∇p0 = δt=0δx=ya in R

3 × R,

∇ · u0 = 0 in R3 × R,

u0(x, t) = 0 for x ∈ R3 and t 0.

The inverse problem considered in this chapter is to image an anomaly D withshear modulus µ inside a background medium of shear modulus µ 6= µ from near-field or far-field measurements of the transient elastic wave u(x, t) (approximatedby u0(x, t)) that is the solution to (2.7) (approximated by (2.8)).

In order to design an accurate and robust algorithm to detect the anomaly Dincorporating the fact that D is of small size of order ε, we will derive an asymptoticexpansion of u0 as ε→ 0. As shown in [3], this scale separation methodology yieldsto accurate imaging algorithms.

2.2.1. Layer potentials for the Stokes system. We begin by reviewingsome basic facts on layer potentials for the Stokes system, which we shall use inthe next subsection. Relevant derivations or proofs of these facts can be found in[77] and [10].

We consider the following modified Stokes system:

(2.9)

(∆ + κ2)v −∇q = 0,

∇ · v = 0.

Here v is the displacement field and q is the pressure. Let ∂i = ∂∂xi

. The funda-

mental tensor Γκ = (Γκij)3i,j=1 and F = (F1, F2, F3) to (2.9) in three dimensions are

given by

(2.10)

Γκij(x) = −δij4π

e√−1κ|x|

|x| − 1

4πκ2∂i∂j

e√−1κ|x| − 1

|x| ,

Fi(x) = − 1

4π

xi|x|3 .


If κ = 0, let

(2.11) Γ0ij(x) = − 1

8π

(δij|x| +

xixj|x|3

).

Then Γ0 = (Γ0ij) together with F is the fundamental tensor for the standard Stokes

system given by ∆v −∇q = 0,

∇ · v = 0.

One can easily see that

(2.12) Γκij(x) = Γ0ij(x) −

δijκ√−1

6π+O(κ2)

uniformly in x as long as |x| is bounded.For a bounded C2-domain D and κ ≥ 0, let

(2.13)

SκD[ϕ](x) :=

∫

∂D

Γκ(x− y)ϕ(y)dσ(y),

QD[ϕ](x) :=

∫

∂D

F(x− y) · ϕ(y) dσ(y),

x ∈ R3

for ϕ = (ϕ1, ϕ2, ϕ3) ∈ L2(∂D)3. When κ = 0, S0D is the single layer potential

for the Stokes system. It is worth emphasizing that SκD[ϕ](x) is a vector whileQD[ϕ](x) is a scalar, and the pair (SκD[ϕ],QD[ϕ]) is a solution to (2.9).

By abuse of notation, let

∂u

∂N= (∇u)N on ∂D.

We define the conormal derivative ∂/∂n (for the Stokes system) on ∂D by

∂v

∂n

∣∣∣∣±

=∂v

∂N

∣∣∣∣±− q∣∣± N

for a pair of solutions (v, q) to (2.9). It is well-known that

(2.14)∂SκD[ϕ]

∂n

∣∣∣∣±

= (±1

2I + (Kκ

D)∗)[ϕ] a.e. on ∂D,

where KκD is the boundary integral operator defined by

(2.15)

KκD[ϕ](x) := p.v.

∫

∂D

[∂

∂N(y)(Γκ(x− y)ϕ(y)) + F(x− y)N(y) · ϕ(y)

]dσ(y)

for almost all x ∈ ∂D and (KκD)∗ is the L2-adjoint operator of K−κ

D :(2.16)

(KκD)∗[ϕ](x) := p.v.

∫

∂D

[∂

∂N(x)(Γκ(x− y)ϕ(y)) + F(x− y) · ϕ(y)N(x)

]dσ(y).

Here p.v. denotes the Cauchy principal value.Let H1(∂D) := ϕ ∈ L2(∂D), ∂ϕ/∂τ ∈ L2(∂D), ∂/∂τ being the tangential

derivative. The operator S0D is bounded from L2(∂D)3 into H1(∂D)3 and invertible

in three dimensions. Moreover, one can see that for κ small

(2.17) ‖SκD[ϕ] − S0D[ϕ]‖H1(∂D) ≤ Cκ‖ϕ‖L2(∂D)


for all ϕ ∈ L2(∂D)3, where C is independent of κ. It is also well-known that thesingular integral operator (K0

D)∗ is bounded on L2(∂D)3. Similarly to (2.17), onecan see that for κ small

‖(K−κD )∗[ϕ] − (K0

D)∗[ϕ]‖L2(∂D) ≤ Cκ‖ϕ‖L2(∂D)

for some constant C independent of κ, which in view of (2.14) yields

(2.18)

∥∥∥∥∥∂(SκD[ϕ])

∂n

∣∣∣∣±− ∂(S0

D[ϕ])

∂n

∣∣∣∣±

∥∥∥∥∥L2(∂D)

≤ Cκ‖ϕ‖L2(∂D).

2.2.2. Derivation of asymptotic expansions. Recall that y is a pointsource in R

3 such that |y − z| ε. Taking the Fourier transform of (2.8) inthe t-variable yields

(2.19)

(∆ +ω2

µ)u0 −

1

µ∇p0 =

1

µδx=y a in R

3 \D,

(∆ +ω2

µ)u0 −

1

µ∇p0 = 0 in D,

u0|+ − u0|− = 0 on ∂D,

(p0|− − p0|+)N + µ∂u0

∂N

∣∣∣+− µ

∂u0

∂N

∣∣∣−

= 0 on ∂D,

∇ · u0 = 0 in R3,

subject to the radiation condition:(2.20)

p0(x) → 0 as r = |x| → +∞,

∂r∇× u0 −√−1

ω√µ∇× u0 = o(

1

r) as r = |x| → +∞ uniformly in

x

|x| ,

where u0 and p0 denote the Fourier transforms of u0 and of p0, respectively. Wesay that (u0, p0) satisfies the radiation condition if (2.20) holds.

Let

U0(x, ω) : =1

µΓ

ω√µ (x− y)a,(2.21)

q0(x) : = F(x− y) · a.(2.22)

Then the pair (U0(x, ω), q0(x)) satisfies

(2.23)

(∆ +ω2

µ)U0 −

1

µ∇q0 =

1

µδx=y a in R

3,

∇ · U0 = 0 in R3.

In view of (2.19) and (2.23), it is natural to expect that u0 converges to U0 as

ε tends to 0. We shall derive an asymptotic expansion for u0 − U0 as ε tends tozero and carefully estimate the dependence of the remainder on the frequency ω.

Let w = u0 − U0 and introduce

p :=

1

µ(p0 − q0) in R

3 \D,

1

µ(p0 − q0) in D.


Then the pair (w, p) satisfies

(2.24)

(∆ +ω2

µ)w −∇p = 0 in R

3 \D,

(∆ +ω2

µ)w −∇p = (

1

µ− 1

µ)(ω2U0 −∇q0) in D,

w|+ − w|− = 0 on ∂D,

µ(∂w

∂N

∣∣∣+− p|+N) − µ(

∂w

∂N

∣∣∣−− p|−N) = (µ− µ)

∂U0

∂Non ∂D,

∇ · w = 0,

(w, p) satisfies the radiation condition.

Therefore, we can represent (w, p) as(2.25)

w(x) =

(1

µ− 1

µ)

∫

D

Γω√µ (x− y)(ω2U0(y) −∇q0(y)) dy + S

ω√µ

D [ϕ](x) in D,

Sω√µ

D [ψ](x) in R3 \D,

and(2.26)

p(x) =

(1

µ− 1

µ)

∫

D

F(x− y) · (ω2U0(y) −∇q0(y)) dy + QD[ϕ](x) in D,

QD[ψ](x) in R3 \D,

where (ϕ,ψ) is the solution to the following system of integral equations(2.27)

Sω√µ

D [ϕ](x) − Sω√µ

D [ψ](x) = (1

µ− 1

µ)

∫

D

Γω√µ (x− y)(ω2U0(y) −∇q0(y)) dy,

µ∂S

ω√µ

D [ϕ]

∂n

∣∣∣+(x) − µ

∂Sω√µ

D [ψ]

∂n

∣∣∣−

(x) = (µ− µ)∂U0

∂N

+(µ

µ− 1)

∂

∂N

∫

D

Γω√µ (x− y)(ω2U0(y) −∇q0(y)) dy

−(µ

µ− 1)

∫

D

F(x− y) · (ω2U0(y) −∇q0(y)) dy N.

In order to prove the unique solvability of (2.27), let us make a change ofvariables: Recalling that D is of the form D = εB + z, we put

(2.28) ϕ(x) = ϕ(εx+ z), x ∈ ∂B,

and define similarly ψ. Then after scaling, (2.27) takes the form

(2.29)

Sεω√

µ

B [ϕ](x) − Sεω√

µ

B [ψ](x) = A(x),

µ∂S

εω√µ

B [ϕ]

∂n

∣∣∣−

(x) − µ∂S

εω√µ

D [ψ]

∂n

∣∣∣+(x) = B(x),

x ∈ ∂B


where A = (A1, A2, A3) and B = (B1, B2, B3) are defined in an obvious way,namely

(2.30) A(x) = ε(1

µ− 1

µ)

∫

B

Γεω√

µ (x− y)(ω2U0(εy + z) −∇q0(εy + z)) dy,

and

B(x) = (µ− µ)∂U0

∂N(εx+ z)

+ ε(µ

µ− 1)

∂

∂N

∫

B

Γεω√

µ (x− y)(ω2U0(εy + z) −∇q0(εy + z)) dy(2.31)

− ε(µ

µ− 1)

∫

D

F(x− y) · (ω2U0(εy + z) −∇q0(εy + z)) dy N(x).

We emphasize that the normal vector N above is that on ∂B.We may rewrite (2.29) as

(2.32) T (ϕ, ψ) = (A,B),

where T is an operator from L2(∂B)3 ×L2(∂B)3 into H1(∂B)3 ×L2(∂B)3 definedby

T (ϕ, ψ) =

Sεω√

µ

B −Sεω√

µ

B

µ∂

∂nS

εω√µ

B |− −µ ∂

∂nS

εω√µ

B |+

ϕ

ψ

.

We then decompose the operator T as

(2.33) T = T0 + Tε,

where

T0(ϕ, ψ) :=

S0B −S0

B

µ∂

∂nS0B |− −µ ∂

∂nS0B |+

ϕ

ψ

,

and Tε = T − T0. Then by (2.17) and (2.18), it follows that

(2.34) ||Tε(ϕ, ψ)||H1(∂B)×L2(∂B) ≤ Cεω(||ϕ||L2(∂B) + ||ψ||L2(∂B)).

Note that S0B is invertible, and since | µ+µ

2(µ−µ) | > 12 , the operator − (µ+µ)

2(µ−µ)I +

(K0B)∗ is invertible as well (see [10]). Thus one can see that T0 is also invertible.

In fact, one can readily check that the solution is explicit.

Lemma 2.1. For (f ,g) ∈ H1(∂B)3 × L2(∂B)3 the solution (ϕ, ψ) = T −10 (f ,g)

is given by

ϕ = ψ + (S0B)−1[f ],

(2.35)

ψ =1

µ− µ

(− (µ+ µ)

2(µ− µ)I + (K0

B)∗)−1

[−µ(−1

2I + (K0

B)∗)(S0B)−1[f ] + g

].

(2.36)


In view of (2.33) and (2.34), one can see that there is ε0 > 0 such that T isinvertible as long as εω ≤ ε0. Moreover T −1 takes the form

(2.37) T −1 = T −10 + E,

where the operator E satisfies

(2.38) ‖E(f ,g)‖L2(∂B)×L2(∂B) ≤ Cεω(‖f‖H1(∂B) + ‖g‖L2(∂B)),

for some constant C independent of ε and ω.

Suppose that εω ≤ ε0 < 1. Let (ϕω, ψω) be the solution to (2.29). Then by(2.37) we have

(ϕω, ψω) = T −10 (A,B) + E(A,B).

In view of (2.30) we have

(2.39) ‖A‖H1(∂B) ≤ Cε(ω2 + 1).

On the other hand, according to (2.31), B can be written as

B(x) = (µ− µ)∇U0(z, ω)N(x) + B1(x),

where B1 satisfies

(2.40) ‖B1‖L2(∂B) ≤ Cε(ω2 + 1).

Therefore, we have

(2.41) (ϕω, ψω) = (µ− µ)T −10

(0, ∇U0(z, ω)N

)+ T −1

0 (A,B1) + E(A,B).

Because of (2.38), (2.39), and (2.40), the last two terms in the above equation areerror terms satisfying

‖T −10 (A,B1) + E(A,B)‖L2(∂B)×L2(∂B) ≤ Cε(ω2 + 1).

We also need to derive asymptotic expansions for ∂ eϕω

∂ω and ∂ eψω

∂ω . By differenti-ating both sides of (2.29) with respect to ω, we obtain

Sεω√

µ

B

[∂ϕω∂ω

](x) − S

εω√µ

B

[∂ψω∂ω

](x) =

∂A(x)

∂ω−∫

∂B

∂

∂ωΓ

εω√µ (x− y)ϕω(y)dσ(y)

+

∫

∂B

∂

∂ωΓ

εω√µ (x− y)ψω(y)dσ(y)(2.42)

and

µ∂

∂nS

εω√µ

B

[∂ϕω∂ω

]∣∣∣∣−

(x) − µ∂

∂nS

εω√µ

B

[∂ψω∂ω

]∣∣∣∣+

(x) =∂B(x)

∂ω

− ∂

∂n

∫

∂B

∂

∂ωΓ

εω√µ (x− y)ϕω(y)dσ(y) +

∂

∂n

∫

∂B

∂

∂ωΓ

εω√µ (x− y)ψω(y)dσ(y)(2.43)

on ∂B.Straightforward computations using (2.10) and (2.30) show that the right-hand

side of the equality in (2.42) is of order ε(ω+ 1) in the H1(∂B)-norm. We can alsoshow using (2.31) that ∂G1

∂ω is also of order ε(ω + 1) in the L2(∂B)-norm. Thus,using the same argument as before, we readily obtain

(2.44) (∂ϕω

∂ω,∂ψω

∂ω) = (µ− µ)T −1

0

(0, ∇(

∂U0

∂ω)(z, ω)N

)+O(ε(ω + 1)),

where the equality holds in L2(∂B)3 × L2(∂B)3.


In view of (2.41) and (2.44), applying Lemma 2.1 (with f = 0) yields thefollowing result.

Proposition 2.2. Let (ϕω, ψω) be the solution to (2.29). There exists ε0 > 0such that if εω < ε0, then the following asymptotic expansions hold:

ϕω =

(−(µ+ µ)

2(µ− µ)I + (K0

B)∗)−1

[∇U0(z, ω)N] +O(ε(ω2 + 1)),(2.45)

ψω =

(−(µ+ µ)

2(µ− µ)I + (K0

B)∗)−1

[∇U0(z, ω)N] +O(ε(ω2 + 1)),(2.46)

and

∂ϕω

∂ω=

(−(µ+ µ)

2(µ− µ)I + (K0

B)∗)−1

[∇ ∂

∂ωU0(z, ω)N] +O(ε(ω + 1)),(2.47)

∂ψω

∂ω=

(−(µ+ µ)

2(µ− µ)I + (K0

B)∗)−1

[∇ ∂

∂ωU0(z, ω)N] +O(ε(ω + 1)),(2.48)

where all the equalities hold in L2(∂B).

We are now ready to derive the inner expansion for w. Let Ω be a domain

containing D and let Ω = 1εΩ − z. After a change of variables, (2.25) and (2.26)

take the forms:(2.49)

w(εx+ z, ω) =

ε2(1

µ− 1

µ)

∫

B

Γεω√

µ (x− y)(ω2U0(εy + z) −∇q0(εy + z)) dy

+εSεω√

µ

B [ϕω](x) in B,

εSεω√

µ

B [ψω](x) in R3 \B,

and(2.50)

p(εx+ z, ω) =

ε(1

µ− 1

µ)

∫

B

F(x− y) · (ω2U0(εy + z) −∇q0(εy + z)) dy

+εQB [ϕω](x) in B,

εQB [ψω](x) in R3 \B.

Since∥∥S

εω√µ

B [ϕω] − S0B [ϕω]

∥∥H1(∂B)

≤ Cεω‖ϕω‖L2(∂B),

we have

w(εx+ z, ω) =

εS0B [ϕω](x) +O(ε2(ω2 + 1)), x ∈ B,

εS0B [ψω](x) +O(ε2(ω + 1)), x ∈ Ω \B.

It then follows from (2.45) and (2.46) that(2.51)

w(εx+ z, ω) = εS0B

(− (µ+ µ)

2(µ− µ)I + (K0

B)∗)−1

[∇U0(z, ω)N](x) +O(ε2(ω2 + 1))

for x ∈ Ω.


On the other hand, we have

∂w

∂ω(εx+ z, ω) =

εSεω√

µ

B

[∂ϕω

∂ω

](x) +O(ε2(ω + 1)), x ∈ B,

εSεω√

µ

B

[∂ψω

∂ω

](x) +O(ε2), x ∈ Ω \B.

Therefore, from (2.47) and (2.48) we obtain that(2.52)

∂w

∂ω(εx+z, ω) = εS0

B

(− (µ+ µ)

2(µ− µ)I + (K0

B)∗)−1

[∇ ∂

∂ωU0(z, ω)N](x)+O(ε2(ω+1))

for x ∈ Ω.Let

v(x) := S0B

(− (µ+ µ)

2(µ− µ)I + (K0

B)∗)−1

[∇U0(z, ω)N](x),

q(x) := QB

(− (µ+ µ)

2(µ− µ)I + (K0

B)∗)−1

[∇U0(z, ω)N](x).

It is easy to check that (v, q) is the solution to

(2.53)

µ∆v −∇q = 0 in R3 \B,

µ∆v −∇q = 0 in B,

v|− − v|+ = 0 on ∂B,

(qN − µ∂v

∂N)

∣∣∣∣−− (qN − µ

∂v

∂N)

∣∣∣∣+

= (µ− µ)∇U0(z, ω)N on ∂B,

∇ · v = 0 in R3,

v(x) → 0 as |x| → +∞,

q(x) → 0 as |x| → +∞.

We finally obtain the following theorem from (2.51) and (2.52).

Theorem 2.3. Let Ω be a small region containing D and let

(2.54) R(x, ω) = u0(x, ω) − U0(x, ω) − εv

(x− z

ε

), x ∈ Ω.


(2.55) R(x, ω) = O(ε2(ω2 + 1)), ∇xR(x, ω) = O(ε(ω2 + 1)), x ∈ Ω.

Moreover,

(2.56)∂R

∂ω(x, ω) = O(ε2(ω + 1)), ∇x

(∂R

∂ω

)(x, ω) = O(ε(ω + 1)), x ∈ Ω.

Note that the estimates for ∇xR in (2.55) and ∇x(∂R∂ω ) in (2.56) can be derived

using (2.49).We now derive the outer expansion of u0. To this end, let us first recall the

notion of the viscous moment tensor (VMT) from [10]. Let (vk`, p), for k, ` = 1, 2, 3,


be the solution to

(2.57)

µ∆vk` −∇p = 0 in R3 \B,

µ∆vk` −∇p = 0 in B,

vk`|− − vk`|+ = 0 on ∂B,

(pN − µ∂vk`∂N

)

∣∣∣∣−− (pN − µ

∂vk`∂N

)

∣∣∣∣+

= 0 on ∂B,

∇ · vk` = 0 in R3,

vk`(x) − xke` +δk`3

3∑

j=1

xjej = O(|x|−2) as |x| → +∞,

p(x) = O(|x|−3) as |x| → +∞.

Here (e1, e2, e3) is the standard basis of R3.

Definition 2.4. The VMT V (µ, µ,B) = (Vijk`)i,j,k,`=1,2,3 is defined by

(2.58) Vijk`(µ, µ,B) := (µ− µ)

∫

B

∇vk`(x) : ∇(xiej) dx,

where : denotes the contraction of two matrices, i.e., A : B =∑3ij=1 aijbij.

Since (u0 − U0, p0 − q0) satisfies(2.59)

(∆ +ω2

µ)(u0 − U0) −

1

µ∇(p0 − q0) = 0 in R

3 \D,

(∆ +ω2

µ)(u0 − U0) −

1

µ∇(p0 − q0) = ω2

(1

µ− 1

µ

)u0 −

(1

µ− 1

µ

)∇p0 in D,

(u0 − U0)∣∣+− (u0 − U0)

∣∣− = 0 on ∂D,

− 1

µ(p0 − q0)

∣∣+N +

∂

∂N(u0 − U0)

∣∣+

= − 1

µ(p0 − q0)

∣∣−N +

∂

∂N(u0 − U0)

∣∣− +

µ− µ

µ

∂u0

∂N

∣∣∣∣−

on ∂D,

∇ · (u0 − U0) = 0 in R3,

together with the radiation condition, the integration of the first equation in (2.59)

against the Green function Γω√µ (x, y) over y ∈ R

3 \D and the divergence theoremgive us the following representation formula:

u0(x) = U0(x) + (µ

µ− 1)

∫

∂D

Γω√µ (x, y)

∂u0

∂N

∣∣∣∣−

(y)dσ(y)

− (1

µ− 1

µ)

∫

D

Γω√µ (x, y)∇p0(y) dy + ω2(

1

µ− 1

µ)

∫

D

Γω√µ (x, y)u0(y) dy.(2.60)

It follows from the inner expansion in Theorem 2.3 that, for y ∈ ∂D,

(2.61)∂u0

∂N(y) =

∂U0

∂N(y) +

∂v

∂N

(y − z

ε

)+O(ε)


and, for x ∈ D,

∇p0(x) = µ4u0 + ω2u0 =µ

ε(4v)

(x− z

ε

)+O(1) =

1

ε(∇q)

(x− z

ε

)+O(1).

(2.62)

Since

µ

∫

∂D

∂u0

∂N

∣∣∣∣−

(y) dσ(y) −∫

D

∇p0(y) dy = −ω2

∫

D

u0(y) dy,

we obtain that for x far away from z, the following outer expansion holds:

u0(x) ≈ U0(x) − ε33∑

i,j,`=1

∂iΓω√µ

`j (x, z)

[(µ

µ− 1)

∫

∂B

(∂U0

∂N(z) +

∂v

∂N

∣∣∣∣−

(ξ)

)

j

ξi dσ(ξ)

(1

µ− 1

µ)

∫

B

∂jq(ξ)ξi dξ

]e`,

where ∂iΓω√µ

`j (x, z) is the differentiation with respect to the x variable and(∂v∂N

)j

is the j-th component of ∂v∂N , which we may further simplify as follows

(2.63)

(u0 − U0)(x)

≈ −ε3( µµ− 1)

3∑

i,j,`=1

[∂iΓ

ω√µ

`j (x, z)

∫

B

∂jvi(ξ) + ∂ivj(ξ) + ∂jU0i(z) + ∂iU0j(z) dξ

]e`.

Here vj denotes the j-th component of v.Since

(2.64) v(ξ) =

3∑

p,q=1

∂qU0(z)pvpq(ξ) −∇U0(z)ξ,

we have(2.65)

(u0 − U0)(x)

≈ −ε3( µµ− 1)

3∑

i,j,`,p,q=1

[∂iΓ

ω√µ

`j (x, z)∂qU0(z)p

∫

B

∂j(vkl)i(ξ) + ∂i(vkl)j(ξ) dξ

]e`.

We have the following theorem for the outer expansion.

Theorem 2.5. Let Ω′ be a compact region away from D, namely dist(Ω′,D) ≥C > 0 for some constant C, and let

(2.66) R(x, ω) = u0(x, ω) − U0(x, ω) +ε3

µ

3∑

i,j,p,q,`=1

Vijkl∂iΓω√µ

`j (x, z)∂qU0(z)pe`.


(2.67) R(x, ω) = O(ε4(ω3 + 1)), x ∈ Ω′.

Moreover,

(2.68)∂R

∂ω(x, ω) = O(ε4(ω2 + 1)), x ∈ Ω′.

2.3. FAR- AND NEAR- FIELD ASYMPTOTICS 39

2.3. Far- and near-field asymptotic formulas in the transient regime

Recall that the inverse Fourier transform, U0, of U0 satisfies

(∂2t − µ∆)U0(x, t) −∇F = δx=yδt=0a in R

3 × R,

∇ · U0 = 0 in R3 × R,

U0(x, t) = 0 for x ∈ R3 and t 0.

For ρ > 0, we define the operator Pρ on tempered distributions by

(2.69) Pρ[ψ](t) =

∫

|ω|≤ρe−

√−1ωtψ(ω) dω,

where ψ denotes the Fourier transform of ψ. The operator Pρ truncates the high-frequency component of ψ.

One can easily show that Pρ[U0] satisfies

(2.70)(∂2t − ∆)Pρ[U0](x, t) −∇Pρ[F ](x− y) = δx=yψρ(t)a in R

3 × R,

∇ · Pρ[U0] = 0 in R3 × R,

where

ψρ(t) :=2 sin ρt

t=

∫

|ω|≤ρe−

√−1ωtdω.

The purpose of this section is to derive and asymptotic expansions for Pρ[u0 −U0](x, t). For doing so, we observe that

(2.71) Pρ[u0](x, t) =

∫

|ω|≤ρe−

√−1ωtu0(x, ω)dω,

where u0 is the solution to (2.19). Therefore, according to Theorem 2.3, we have

Pρ[u0−U0](x, t)−ε3∑

p,q=1

∂qPρ[U0](z, t)p[vpq(x)−xpeq] =

∫

|ω|≤ρe−

√−1ωtR(x, ω)dω.

Suppose that |t| ≥ c0 for some positive number c0 (c0 is of order the distancebetween y and z). Then, integrating by parts gives∣∣∣∣∣

∫

|ω|≤ρe−


∣∣∣∣∣ =∣∣∣∣∣1

t

∫

|ω|≤ρ

d

dωe−


∣∣∣∣∣

≤ 1

|t| (|R(x, ρ)| + |R(x,−ρ)|) +

∫

|ω|≤ρ

∣∣∣∣∂

∂ωR(x, ω)

∣∣∣∣ dω

≤ Cε2ρ2.

Since

ε

3∑

p,q=1

∂qPρ[U0](z, t)p[vpq(x) − xpeq] = O(ερ),

we arrive at the following theorem.

Theorem 2.6. Suppose that ρ = O(ε−α) for some α < 1. Then

Pρ[u0 − U0](x, t) = ε

3∑

p,q=1

∂qPρ[U0](z, t)p[vpq(x) − xpeq] +O(ε2(1−α)).


We now derive a far-field asymptotic expansion for Pρ[u0−U0]. Let G∞(x, y, t)

be the inverse Fourier transform of Γω√µ (x, y). Note that G∞ is the limit of G given

by (2.5) as√λ+ 2µ→ +∞. It then follows that

Pρ[G∞](x, y, t) =

∫

|ω|≤ρe−

√−1ωtΓ

ω√µ (x, y)dω

=1

4π

3γiγj − δijr3

[φρ(t) − φρ(t−

r√µ

)

]− 1

4πµ

γiγj − δijr

ψρ(t−r√µ

),(2.72)

where φρ(t) :=∫ t0ψρ(s)ds.

From Theorem 2.5, we get∫

|ω|≤ρe−

√−1ωt(u0(x, ω) − U0(x, ω)) dω

= −ε3

µ

∫

|ω|≤ρe−

√−1ωt

3∑

i,j,p,q,`=1

Vijpq∂iΓω√µ

`j (x, z)∂qU0(z)pe`

dω

+

∫

|ω|≤ρe−

√−1ωtR(x, ω) dω,

where the remainder is estimated by∫

|ω|≤ρe−

√−1ωtR(x, ω) dω = O(ε4(1−

34α)).

Since

∫

|ω|≤ρe−

√−1ωt

3∑

i,j,p,q,`=1

Vijpq∂iΓω√µ

`j (x, z)∂qU0(z)pe`

dω

= µ−1

∫

|ω|≤ρe−

√−1ωt

3∑

i,j,p,q,k,`=1

Vijpq∂iΓω√µ

`j (x, z)∂qΓω√µ

pk (z, y)ake`

dω

= µ−1

∫

R

3∑

i,j,p,q,k,`=1

Vijpq∂iPρ[G∞]`j(x, z, t− τ)∂qPρ[G∞]pk(z, y, τ)ake`

dτ,

the following theorem holds.

Theorem 2.7. Let U0(x, ω) := 1µΓ

ω√µ (x − y)a. Suppose that ρ = O(ε−α) for

some α < 1. Then for |x− z| ≥ C > 0, the following far-field expansion holds(2.73)

Pρ[u0 − U0](x, t)

= − ε3

µ2

∫

R

3∑

i,j,p,q,k,`=1

Vijpq∂iPρ[G∞]`j(x, z, t− τ)∂qPρ[G∞]pk(z, y, τ)ake`

dτ

+O(ε4(1−34α)).

Note that if we plug (2.72) in the far-field formula (2.73) then we can see that,unlike the acoustic case investigated in [8], the perturbation Pρ[u0 − U0](x, t) canbe seen not only as a polarized wave emitted from the anomaly but it contains,

2.4. ASYMPTOTIC IMAGING 41

because of the term (1/r3)φρ(t) in (2.72), a near-field like term which does notpropagate.

2.4. Asymptotic imaging

2.4.1. Far-field imaging: time-reversal. We present a time-reversal tech-nique for detecting the location z of the anomaly from measurements of the per-turbations at x away from the location z. As in the acoustic case, the main ideais to take advantage of the reversibility of the elastic wave equation in a non-viscous medium in order to back-propagate signals to the sources that emittedthem [28, 57].

Let S be a sphere englobing the anomaly D. Consider, for simplicity, theharmonic regime, we get

∫

S

[∂Γ

ω√µ

∂n(x, z)Γ

ω√µ (x, y) − Γ

ω√µ (x, z)

∂Γω√µ

∂n(x, y)

]dσ(x) = 2

√−1=mΓ

ω√µ (y, z),

for y ∈ Ω, and therefore, for w(x) := u0(x, ω) − U0(x, ω), it follows that

∫

S

[∂w

∂n(x, ω)Γ

ω√µ (x, z) − w(x, ω)

∂Γω√µ

∂n(x, z)

]dσ(x)

= 2√−1

ε3

µ∇U0(z, ω)V (µ, µ,B)∇z=mΓ

ω√µ (y, z) +O(ε4ω3),

if ω > 1.This shows that the anti-derivative of time-reversal perturbation focuses on the

location of the anomaly with an anisotropic focal spot. Because of the structure

of the Green function Γω√µ (y, z), time-reversing the perturbation gives birth to a

near-field like effect. Moreover, the resolution limit depends on the direction. It is,unlike the acoustic case, anisotropic. These interesting findings were experimentallyobserved and first reported in [43]. Our asymptotic formula (2.73) clearly explainsthem.

2.4.2. Near-field imaging: optimization approach. Set Ω to be a windowcontaining the anomaly D. As in Chapter 1, Theorem 2.6 suggests to reconstructthe shape and the shear modulus of the elastic inclusion D by minimizing thefollowing functional:

∫ T+∆T

T−∆T

||Pρ[u0 − U0](x, t) − ε

3∑

p,q=1

∂qPρ[U0](z, t)p[vpq(x) − xpeq]||2L2(Ω),

where T = |y − z|/√µ is the arrival time and ∆T is a window time. One can adda total variation regularization term.

The choice of the space and time window sizes are critical as observed in [9] forthe time-harmonic regime. If they are too large, then noisy images are obtained. Ifthey are too small, then resolution is poor. The optimal window sizes are relatedto the signal-to-noise ratio of the recorded near-field measurements. They expressthe trade-off between resolution and stability.



In this chapter we have rigorously establish asymptotic expansions of near-and far-field measurements of the transient elastic wave induced by a small elasticanomaly. We have proved that, after truncation of the high-frequency component,the perturbation due to the anomaly can be seen not only as a polarized waveemitted from the anomaly but it contains unlike the acoustic case a near-fieldlike term which does not propagate. We have also shown that time-reversing thisperturbation gives birth to a near-field like effect. Moreover, the resolution limit isanisotropic. We have then explained the experimental findings reported in [43].

In this chapter we have only considered a purely quasi-incompressible elasticitymodel. In Chapter 4, we will consider the problem of reconstructing a small anomalyin a viscoelastic medium from wavefield measurements. The Voigt model his acommon model to describe the viscoelastic properties of tissues. Catheline et al.[42] have shown that this model is well adapted to describe the viscoelastic responseof tissues to low-frequency excitations. Expressing the ideal elastic field without anyviscous effect in terms of the measured field in a viscous medium, we will generalizethe methods described here to recover the viscoelastic and geometric properties ofan anomaly from wavefield measurements.

CHAPTER 3

Transient imaging with limited-view data

Abstract. We consider for the wave equation the inverse problem of identi-

fying locations of point sources and dipoles from limited-view data. Using as

weights particular background solutions constructed by the geometrical con-trol method, we recover Kirchhoff-, back-propagation-, MUSIC-, and arrival

time-type algorithms by appropriately averaging limited-view data. We show

that if one can construct accurately the geometric control, then one can per-

form imaging with the same resolution using limited-view as using full-view

data.

3.1. Introduction

In Chapter 1, we have investigated the imaging of small anomalies using tran-sient wave boundary measurements; see also the recent works [5, 7]. Different ap-proaches for locating them and reconstructing some information about their sizesand physical parameters have been designed. The detection algorithms make useof complete boundary measurements. They are of Kirchhoff-, back-propagation,MUSIC-, and arrival time-types. The resolution of those algorithms in the time-harmonic domain is finite. It is essentially of order one-half the wavelength. See,for instance, [3].

In this work, we extend those algorithms to the case with limited-view mea-surements. For simplicity, we model here the small anomalies as point sources ordipoles. We refer the reader to Chapter 1 and [5, 7] for rigorous derivations of theseapproximate models and their higher-order corrections. It is worth mentioning thatin order to model a small anomaly as a point source or a dipole, one has to truncatethe high-frequency component of the transient wave reflected by the anomaly.

By using the geometrical control method [29], we show how to recover allthe classical algorithms that have been used to image point sources and dipolelocations. Our main finding in this chapter is that if one can construct accuratelythe geometric control then one can perform imaging with the same resolution usingpartial data as using complete data. Our algorithms apply equally well to the case ofmany source points or dipole locations and are robust with respect to perturbationsof the boundary. This is quite important in real experiments since one does notnecessarily know the non-accessible part of the boundary with good accuracy.

The chapter is organized as follows. In Section 3.2 we provide a key identitybased on the averaging of the limited-view data, using weights constructed by thegeometrical control method. Section 3.3 is devoted to developing, for differentchoices of weights, Kirchhoff-, back-propagation-, MUSIC-, and arrival time-typealgorithms for transient imaging with limited-view data. In Section 3.4 we discusspotential applications of the method in emerging biomedical imaging. In Section 3.5

43

44 3. TRANSIENT IMAGING WITH LIMITED-VIEW DATA

we present results of numerical experiments and comparisons among the proposedalgorithms.

3.2. Geometric control

The basic model to be considered in this chapter is the following wave equation:

(3.1)∂2p

∂t2(x, t) − c2∆p(x, t) = 0, x ∈ Ω, t ∈]0, T [,

for some final observation time T , with the Dirichlet boundary conditions

(3.2) p(x, t) = 0 on ∂Ω×]0, T [,

the initial conditions

(3.3) p(x, t)|t=0 = 0 in Ω,

and

(3.4) ∂tp(x, t)|t=0 = δx=z or ∂tp(x, t)|t=0 = m0 · ∇δx=z in Ω.

Here c is the acoustic speed in Ω which we assume to be constant, and m0 is aconstant nonzero vector. We suppose that T is large enough so that

(3.5) T >diam(Ω)

c.

The purpose of this chapter is to design efficient algorithms for reconstructingthe location z from boundary measurements of ∂p

∂ν on Γ×]0, T [, where Γ ⊂ ∂Ω.Suppose that T and Γ are such that they geometrically control Ω, which roughly

means that every geometrical optic ray, starting at any point x ∈ Ω, at time t = 0,hits Γ before time T at a nondiffractive point; see [29, 78]. Let β ∈ C∞

0 (Ω) be acutoff function such that β(x) ≡ 1 in a sub-domain Ω′ of Ω, which contains thesource point z.

For a given function w which will be specified later, we construct by the geo-metrical control method a function v(x, t) satisfying

(3.6)∂2v

∂t2− c2∆v = 0 in Ω×]0, T [,

with the initial condition

(3.7) v(x, 0) = c2β(x)w(x), ∂tv(x, 0) = 0,

the boundary condition v = 0 on ∂Ω \ Γ, and the final conditions

(3.8) v|t=T =∂v

∂t

∣∣∣t=T

= 0 in Ω.

Let

(3.9) gw(x, t) := v(x, t) on Γ×]0, T [.

Multiplying (3.1) by v and integrating over Ω × [0, T ] lead to the following keyidentity of this chapter:

(3.10)

∫ T

0

∫

Γ

∂p

∂ν(x, t)gw(x, t) dσ(x) dt = w(z) or −m0 · ∇w(z).

3.3. IMAGING ALGORITHMS 45

Note that the probe function constructed in [5] corresponds to one of the fol-lowing choices for w in Ω:

(3.11) w(x) :=δ(τ − |x−y|

c

)

4π|x− y| in three dimensions

or

(3.12) w(x) := δ

(τ − 1

cθ · x

)in two dimensions,

where θ is a unit vector.The reader is referred, for instance, to [25, 106, 68] for numerical investigations

of the geometrical control method.

3.3. Imaging algorithms

In this section, we only consider the initial condition ∂tp(x, t)|t=0 = δx=z in Ω.One can treat the case of the initial data ∂tp(x, t)|t=0 = m0 · ∇δx=z in the exactlysame way. Using the functions v constructed by the geometrical control methodwith different choices of initial data w, one recovers several classical algorithms forimaging point sources. For simplicity, we only consider a single point source, butthe derived algorithms are efficient for locating multiple sources as well. The readeris referred to [48] for a review on source localization methods.

3.3.1. Kirchhoff algorithm. Let y ∈ Rd \ Ω, d = 2, 3, and ω ∈ R. Set

w(x) = e√−1ω|x−y|, x ∈ Ω.

Then, for a given search point zS in Ω, we have from (3.10)∫

R

e−√−1ω|zS−y|

∫ T

0

∫

Γ

∂p

∂ν(x, t)gw(x, t) dσ(x) dt dω =

∫

R

e−√−1ω(|zS−y|−|z−y|) dω

= δ|zS−y|−|z−y|=0,

where δ is the Dirac mass. Taking a (virtual) planar array of receivers y outside Ωyields then a Kirchhoff-type algorithm for finding z.

In fact, let ωk, k = 1, . . . ,K, be a set of frequencies and let y1, . . . , yN , be aset of virtual receivers. To find the location z one maximizes over zS the followingimaging functional:

IKI(zS) :=

1

K<e

∑

ωk

∑

yn

e−√−1ω|zS−yn|

∫ T

0

∫

Γ

∂p

∂ν(x, t)gwk,n

(x, t) dσ(x) dt,

where wk,n(x) = e√−1ωk|x−yn|.

3.3.2. Back-propagation algorithm. If one takes w to be a plane wave:

w(x) = e√−1ωθ·x, θ ∈ Sd−1,

where Sd−1 is the unit sphere in Rd, then one computes for a given search point

zS ∈ Ω,∫

Sd−1

e−√−1ωθ·zS

∫ T

0

∫

Γ

∂p

∂ν(x, t)gw(x, t) dσ(x) dt dσ(θ) =

∫

Sd−1

e√−1ωθ·(z−zS) dσ(θ).


But ∫

Sd−1

e√−1ωθ·(z−zS) dσ(θ) =

j0(ω|z − zS |) for d = 3,

J0(ω|z − zS |) for d = 2,

where j0 is the spherical Bessel function of order zero and J0 is the Bessel functionof the first kind and of order zero.

This is a back-propagation algorithm. Let θ1, . . . , θN , be a discretization of theunit sphere Sd−1. One plots at each point zS in the search domain the followingimaging functional:

IBP(zS) :=1

N<e

∑

θn

e−√−1ωθn·zS

∫ T

0

∫

Γ

∂p

∂ν(x, t)gwn

(x, t) dσ(x) dt,

where wn(x) = e√−1ωθn·x. The resulting plot will have a large peak at z. Note that

the higher the frequency ω is, the better is the resolution. However, high frequencyoscillations cause numerical instabilities. There is a trade-off between resolutionand stability.

3.3.3. MUSIC algorithm. Take

w(x) = e√−1ω(θ+θ′)·x, θ, θ′ ∈ Sd−1.

It follows from (3.10) that∫ T

0

∫

Γ

∂p

∂ν(x, t)gw(x, t) dσ(x) dt = e

√−1ω(θ+θ′)·z.

Therefore, one can design a MUSIC-type algorithm for locating z. For doing so, letθ1, . . . , θN be N unit vectors in R

d. Define the matrix A = (Ann′)Nn,n′=1 by

Ann′ :=

∫ T

0

∫

Γ

∂p

∂ν(x, t)gwn,n′ (x, t) dσ(x) dt,

with

wn,n′(x) = e√−1ω(θn+θn′ )·x.

Let P be the orthogonal projection onto the range of A. Given any point zS in thesearch domain form the vector

h(zS) := (e√−1ωθ1·zS

, . . . , e√−1ωθN ·zS

)T ,

where T denotes the transpose. Then plot the MUSIC imaging functional:

IMU(zS) :=1

||(I − P )h(zS)|| .

The resulting plot will have a large peak at z. Again, the higher the frequency ωis, the better is the resolution.

3.3.4. Arrival time and time-delay of arrival algorithms. Taking w tobe a distance function,

w(x) = |y − x|,to a virtual receiver y on a planar array outside Ω yields arrival-time and time-delayof arrival algorithms. In fact, we have

∫ T

0

∫

Γ

∂p

∂ν(x, t)gw(x, t) dσ(x) dt = |y − z|.

3.4. APPLICATIONS TO EMERGING BIOMEDICAL IMAGING 47

Let y1, . . . , yN be N receivers and compute

rn :=

∫ T

0

∫

Γ

∂p

∂ν(x, t)gwn

(x, t) dσ(x) dt,

with wn(x) = |yn−x|. Then, the point z can be found as the intersection of spheresof centers yn and radii rn.

Using time-of-arrival differences instead of arrival times would improve therobustness of the algorithm. Introduce the time-of-arrival difference, tn,n′ , betweenthe receiver yn and yn′ as follows:

tn,n′ :=

∫ T

0

∫

Γ

∂p

∂ν(x, t)(gwn

− gwn′ )(x, t) dσ(x) dt.

At least N = 4 sources are required to locate z. The location z can be found as theintersection of three sets of hyperboloids. See, for instance, [40, 104, 96, 47, 66,32, 48].

3.4. Applications to emerging biomedical imaging

In this section we show how to apply the designed algorithms to emergingbiomedical imaging. Of particular interest are radiation force imaging, magneto-acoustic current imaging, and photo-acoustic imaging.

3.4.1. Radiation force imaging. As it has been said Chapter 1, in radiationforce imaging, one generates vibrations inside the organ, and acquires a spatio-temporal sequence of the propagation of the induced transient wave to estimate thelocation and the viscoelastic parameters of a small anomaly inside the medium.

Let z be the location of the anomaly. Let Ω be a large ball englobing theanomaly. In the far-field, the problem, roughly speaking, reduces to finding thelocation of the anomaly from measurements of the pressure p on ∂Ω×]0, T [, thatis, the solution to (3.1) with the initial conditions

(3.13) p(x, t)|t=0 = 0 and ∂tp(x, t)|t=0 = m0 · ∇δx=z in Ω.

A time-reversal technique can be designed to locate the anomaly. Suppose thatone is able to measure p and its normal derivative at any point x on ∂Ω. If both pand its normal derivative on ∂Ω are time-reversed and emitted from ∂Ω, then thetime-reversed wave travels back to the location z of the anomaly. See Chapter 1.

Suppose now that the measurements of p and its normal derivative are onlydone on the part Γ of ∂Ω. Note first that

∂p

∂ν|∂Ω×]0,T [ = ΛDtN[p|∂Ω×]0,T [],

where ΛDtN is the Dirichlet-to-Neumann operator for the wave equation in R3 \Ω.

For any function v satisfying (3.6), (3.7), and (3.8), integrating by parts yields∫ T

0

∫

∂Ω

p(x, t)(Λ∗DtN[v] +

∂v

∂ν)(x, t) dσ(x) dt = m0 · ∇w(z),

where Λ∗DtN denotes the adjoint of ΛDtN. Next, constructing by the geometrical

control method, gw such that v satisfies (3.6), (3.7), and (3.8), together with theboundary condition

Λ∗DtN[v] +

∂v

∂ν=

0 on ∂Ω \ Γ ×]0, T [

gw on Γ×]0, T [,


one obtains ∫ T

0

∫

Γ

p(x, t)gw(x, t) dσ(x) dt = m0 · ∇w(z).

Making similar choices for w to those in the previous section provide differentalgorithms for locating the anomaly.

3.4.2. Magneto-acoustic current imaging. In magneto-acoustic currentimaging, one detects a pressure signal created in the presence of a magnetic fieldby electrically active tissues [70, 90, 91]. In the presence of an externally appliedmagnetic field, biological action currents, arising from active nerve or muscle fibers,experience a Lorentz force. The resulting pressure or tissue displacement containsinformation about the action current distribution.

Let z ∈ Ω be the location of an electric dipole, which represents an active nerveor muscle fiber, with strength c. The wave equation governing the induced pressuredistribution p is (3.1), with the boundary condition (3.2), the initial conditions(3.3), and

(3.14) ∂tp(x, t)|t=0 = δx=z in Ω.

The algorithms constructed in the previous section apply immediately to finding zfrom partial boundary measurements of the normal derivative of p.

3.4.3. Photo-acoustic imaging. The photo-acoustic effect refers to the gen-eration of acoustic waves by the absorption of optical energy [105, 56]. In photo-acoustic imaging, energy absorption causes thermo-elastic expansion of the tissue,which in turn leads to propagation of a pressure wave. This signal is measured bytransducers distributed on the boundary of the organ, which is in turn used forimaging optical properties of the organ. Mathematically, the pressure p satisfies(3.1) with the boundary condition (3.2) and the initial conditions

(3.15) p(x, t)|t=0 = aδx=z in Ω,

and

(3.16) ∂tp(x, t)|t=0 = 0 in Ω.

Here a is the absorbed energy.Construct by the geometrical control method a function v(x, t) satisfying (3.6),

the initial condition (3.7), the boundary condition v = 0 on ∂Ω \ Γ, and the finalconditions (3.8). Choosing w as in Section 3.4 yields different detection algorithms.


To test the geometrical control imaging approach, we implemented numericalsimulations of both the forward problem, the wave equation (3.1)-(3.4), and theinverse problem where we compute the geometrical control function (3.6)-(3.9) andimplement the inversion algorithms of Section 3.3.

To simulate the wave equation, we used a standard P1-finite elements discretiza-tion in space and a finite difference scheme in time. For time-cost considerations,we settled with an explicit (leap-frog) scheme along with the use of mass lumping(row-sum technique).

The method we present here has been implemented and tested on various typesof two-dimensional meshes. We will present results obtained on three different setsof meshes (see Figure 3.1 and Table 3.1):


Set name Coarse mesh Fine mesh# of nodes # of elements h # of nodes # of elements 2h

squareReg0 36 50 0.2 121 200 0.1squareReg2 441 800 0.05 1681 3200 0.025

circle 270 490 0.0672 1029 1960 0.0336Table 3.1. Geometries and meshes.

• squareReg0 and squareReg2 are regular meshes of the unit square [−0.5 0.5]2.• circle are unstructured meshes of the unit disc.

−0.5 0 0.5−0.5

0

0.5coarse

−0.5 0 0.5−0.5

0

0.5fine

−0.5 0 0.5−0.5

0

0.5coarse

−0.5 0 0.5−0.5

0

0.5fine

−1 0 1−1

−0.5

0

0.5

1coarse

−1 0 1−1

−0.5

0

0.5

1fine

Meshes squareReg0

Meshes circle

Meshes squareReg2

Figure 3.1. The coarse and fine meshes used on the square andcircular geometries.

For computation of imaging functionals of Kirchhoff-, back-propagation-, andMUSIC-types, one has to be very careful with the spatial frequency ω. One hasto make sure that the function w(x;ω) is accurately represented on the meshes weuse. This imposes strict limitations on the range of frequencies that can be used.


Finally, the considered initial conditions for the simulated measurements arep(x, 0) = 0 and ∂p

∂t (x, 0) = δh(x0), where δh is a Gaussian approximation of theDirac distribution and x0 = [0.21 − 0.17] (see Figure 3.2).

Figure 3.2. Initial time derivative, for the three geometries, usedfor the simulated measures.

To illustrate the performance of our approach with regards to limiting the view,we applied the algorithm to both a full and a partial view setting.

For the square medium, we assumed measurements were taken only on twoadjacent edges - this corresponds to the theoretical (and practical) limit that stillensures geometric controllability. For the circular medium, we assumed measure-ments between angles π

4 and 3π2 , as shown in Figure 3.4.

Figure 3.3. Limited-view observation boundaries for square and disc.

Before presenting the numerical results, we describe the numerical method usedfor computing the geometrical control, which is based on the Hilbert UniquenessMethod (HUM) of Lions.


3.5.1. Geometrical control: HUM using conjugate gradient iterationon a bi-grid mesh. The solution gw of (3.6)-(3.9) has been shown to be uniqueprovided that T and the control boundary Γ geometrically control Ω [29]. A system-atic and constructive method for computing such a control is given by the HilbertUniqueness Method (HUM) of Lions [79]. A detailed study of the algorithm can befound in [59], [25], and [106]. The method applies a conjugate gradient algorithmas follows:

• Let e0, e1 ∈ H10(Ω) × L2(Ω), where H1

0(Ω) is the standard Sobolev spacewith zero boundary values;

• Solve forwards on (0, T ) the wave equation

(3.17)

∂2φ

∂t2(x, t) − c2∆φ(x, t) = 0,

φ(x, t) = 0 on ∂Ω,

φ(x, 0) = e0(x),∂φ

∂t(x, 0) = e1(x);

• Solve backwards the wave equation

(3.18)

∂2ψ

∂t2(x, t) − c2∆ψ(x, t) = 0,

ψ(x, t) =

0 on ∂Ω\Γ,∂φ∂ν (x, t) on Γ,

ψ(x, T ) = 0,∂ψ

∂t(x, T ) = 0;

• Set

(3.19) Λ(e0, e1) =

∂ψ

∂t(x, 0),−ψ(x, 0)

;

• The solution v of (3.6)-(3.8) can be identified with ψ when

Λ(e0, e1) =0,−c2β(x)w(x)

and gw(x, t) = ψ(x, t) on Γ.

Remark 3.1. In the case where the initial condition is a pressure field (e.g.,

photo-acoustics) p(x, 0) = p0(x),∂p∂t (x, 0) = 0, we need to have v(x, 0) = 0,

∂v∂t (x, 0) = c2β(x)w(x). This can be easily obtained by solving : Λ(e0, e1) =c2β(x)w(x), 0

.

To proceed, we used a conjugate gradient algorithm on a discretized version Λhof the operator defined in (3.19), where we solve the wave equation using the finite-element finite-difference discretization described previously. To deal with unwantedeffects linked with high spatial frequencies, we used a bi-grid method of Glowinski[59] based on a fine mesh with discretization length h and a coarse mesh withlength 2h. The wave equation is solved on the fine mesh and the residuals of Λhare computed after projection onto the coarse mesh.

Let us define I2hh and Ih2h to be the projectors from the fine mesh to the coarse

mesh and vice versa. The conjugate gradient algorithm is now as follows:

• Let e00, e01 be given initial guesses on the coarse mesh;

• Solve numerically (3.17) forwards with initial conditions Ih2he00, I

h2he

01 and

solve (3.18) backwards, both on the fine grid;


• Compute the initial residuals g0 = g00 , g

01 on the coarse grid as follows:

−∆g00 = I2h

h

ψ1 − ψ−1

2∆t− I2h

h u1 in Ω,

g00 = 0 on ∂Ω,

and

g01 = ψ0 − I2h

h u0;

• If the norm of the residuals

‖g00 , g

01‖2

h =

∫

Th

|g01 |2 + |∇g0

0 |2

is small enough, we have our solution, else we set the first search directionw0 = g0 and start the conjugate gradient;

• Suppose we know ek = ek0 , ek1, gk = gk0 , gk1 and wk = wk0 , wk1;• Solve numerically (3.17) forwards with initial conditions Ih2hw

k0 , Ih2hw

k1 and

solve (3.18) backwards both on the fine grid;• Compute the remaining residuals ξk = ξk0 , ξk1 on the coarse grid as

follows:

−∆hξk0 = I2h

h

ψ1 − ψ−1

2∆t,

ξk0 = 0 on ∂Ω,

and

ξk1 = ψ0;

• Calculate the length of the step in the wk direction

ρk =‖gk‖h

〈ξk, wk〉h,

where 〈ξk, wk〉h =

∫

Th

∇ξk0∇wk0 + ξk1wk1 ;

• Update the quantities

ek+1 = ek − ρkwk,

gk+1 = gk − ρkξk;

• If ‖gk+1‖h is small, then ek+1 is our solution, else compute

γk =‖gk+1‖h‖gk‖h

,

and set the new descent direction

wk+1 = gk+1 + γkwk.

Remark 3.2 (Remarks on the numerical convergence). The numerical proce-dure described in the previous section has been proved to converge in the case offinite difference method on the unit square [68]. This result can be easily extendedin the case of a finite element method on a regular mesh. Convergence results formore general meshes are not available yet. They will be the subject of a future study.


3.5.2. Reconstruction results. We present here some results obtained byalgorithms presented in Section 3.4. For each algorithm we will consider both thefull view and the partial view cases.

• Kirchhoff algorithm. We limited ourselves to the frequency range :W = [−ωmax, ωmax] with a step-size ∆ω = ωmax/nω where ωmax and nωdepend on the mesh coarseness.

For time considerations we chose a reduced array of three virtualreceivers

– Y = [0.6 − 0.6], [0.6 0], [0.6 0.6] for the square medium.– Y = [1 − 1], [1 0], [1 1] for the circular medium.

We compute and represent the function IKI(zS) for zS on the fine

mesh. The estimated position is at the maximum of IKI(zS). Recon-

struction results are given in Figure 3.4.• Back-propagation algorithm. We chose frequencies well represented

on the mesh (ω = 9 for squareReg0, ω = 30 for squareReg2 and ω = 20for circle) and a 30-point discretization of the unit circle for θ.

We compute and represent the function IBP(zS) for zS on the finemesh. The estimated position is at the maximum of IBP(zS). Results aregiven in Figure 3.5.

• Arrival-time algorithm. We considered minimal arrays of two virtualreceivers Y = [0 0.6]; [0.6 0] for the square medium. For each receiverwe computed the value of rk = d(x0, yk), where x0 is the position ofthe source and yk the position of the receiver. We represent the circlesC(yk, rk) and their intersections. Results are given in Figure 3.6.

• MUSIC algorithm. Working with the same parameters, we computeand represent the function IMU(zS) for zS on the fine mesh. The esti-mated position is at the maximum of IMU(zS). Reconstruction resultsare given in Figure 3.7.

In Table 3.2 we give the estimations xest of the source location x0 = [0.21−0.17]for each algorithm, and the error d(x0, xest). For comparison, we give hmin, thesmallest distance between 2 points in the fine mesh.

3.5.3. Case of multiple sources. Except for the arrival-time algorithm, allthe methods presented in this chapter are well-suited for identifying several point-like sources. To illustrate this, we simulated measurements on squareReg2 withthree sources located at [0.21 − 0.17], [−0.22 − 0.3] and [0.05 0.27].

• We applied the Kirchhoff imaging algorithm with a different set of virtualreceivers:

Y = [0.6 0], [0.6 0.6], [0 0.6], [−0.6 0.6], [−0.6 0] .The reason for taking more virtual receivers is that Kirchhoff works on in-tersecting circles centered at the receivers and passing through the sources.Too few receivers can generate false positives. Results are given in Figure3.10.

• We ran the back-propagation and MUSIC algorithms with exactly thesame parameters as previously. Results are given in Figures 3.11 and 3.12respectively.


Algorithm Mesh View xest hmin d(x0, xest)Kirchhoff squareReg0 Full [0.2 -0.15] 0.1 0.0224

Partial [0.2 -0.15] 0.0224squareReg2 Full [0.2 -0.175] 0.025 0.0112

Partial [0.2 -0.175] 0.0112circle Full [0.1949 -0.1619] 0.0336 0.0171

Partial [0.1949 -0.1619] 0.0171Back-propagation squareReg0 Full [0.2 -0.15] 0.1 0.0224



Partial [0.1949 -0.1619] 0.0171Arrival time squareReg0 Full [0.1877 -0.1433] 0.1 0.0348



Partial [0.1790 -0.2119] 0.0522MUSIC squareReg0 Full [0.15 -0.2] 0.1 0.0671



Partial [0.2416 -0.0974] 0.0792Table 3.2. Numerical results for localization of the source at x0 =[0.21,−0.17] using four algorithms and three geometries.

3.5.4. Boundary perturbation. In real experiments, one does not neces-sarily know the uncontrolled part of the boundary with good accuracy. A majorconcern for real applications of the method is thus its robustness with respect toperturbations of the boundary.

We tested our algorithms by perturbing the boundary nodes outwards

xi,perturbed = xi + εUnxi,

where ε is an amplitude factor, U is a uniform random variable in [0 1] and nxiis

the outward normal at the point xi. We simulated measurements on the perturbedmesh, which is then supposed unknown since we computed the geometric controlon the unperturbed mesh.

To illustrate the results, we used squareReg2 with three levels of perturbation,ε = 0.01, 0.025 and 0.05 (see Figure 3.5.4) and the same initial condition as before,that is a Dirac approximation located at [0.21 − 0.17].

We give the results, with the three perturbations, for the Kirchhoff (Figure3.13), the back-propagation (Figure 3.14) and the arrival-time (Figure 3.15) algo-rithms. Modifying the mesh as we did generates smaller elements and thus changesthe CFL condition for the wave-equation solver. Computation time becomes too


Figure 3.4. Kirchhoff results for the geometries of Table 3.1- from top to bottom: squareReg0, squareReg2, circle. The(black/white) x denotes the (numerical/theoretical) center of thesource.

expensive for the MUSIC algorithm. For this reason we do not present MUSICresults here.

As expected the estimation of the source position deteriorates as we increasethe boundary uncertainty. The errors are summarized in Table 3.3.


Figure 3.5. Back-propagation results for the geometries of Table3.1 - from top to bottom: squareReg0, squareReg2, circle. The(black/white) x denotes the (numerical/theoretical) center of thesource.


In this chapter we have constructed Kirchhoff-, back-propagation-, MUSIC-,and arrival time-type algorithms for imaging point sources and dipoles from limited-view data. Our approach is based on averaging of the limited-view data, using


−1 −0.5 0 0.5 1

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

virtual receiversreal acoustic receiverstrue position of the sourceintersections of the circles

0.17 0.18 0.19 0.2 0.21 0.22 0.23 0.24

−0.2

−0.19

−0.18

−0.17

−0.16

−0.15

Figure 3.6. Example of arrival time results for squareReg2 geometry.

weights constructed by the geometrical control method. It is quite robust withrespect to perturbations of the non-accessible part of the boundary. We have shownthat if one can construct accurately the geometric control then one can performimaging with the same resolution using partial data as using complete data.


Figure 3.7. MUSIC results for the geometries of Table 3.1 - fromtop to bottom: sqReg0, sqReg2, circle. The (black/white) x de-notes the (numerical/theoretical) center of the source.


Figure 3.8. Initial time derivative for the case of multiple sources.

−0.5 0 0.5

−0.4

−0.2

0

0.2

0.4

0.6

−0.5 0 0.5

−0.4

−0.2

0

0.2

0.4

0.6

−0.5 0 0.5

−0.4

−0.2

0

0.2

0.4

0.6

Figure 3.9. Perturbation of the mesh for ε = 0.01, 0.025 and 0.05.


Figure 3.10. Kirchhoff results for the geometry sqReg2 with sev-eral inclusions.

Figure 3.11. Back-propagation results for the geometry sqReg2with several inclusions.


Figure 3.12. MUSIC results for the geometry sqReg2 with sev-eral inclusions.

Figure 3.13. Kirchhoff results for the geometry sqReg2 with per-turbed boundary (from left-to-right,ε = 0.01, 0.025 and 0.05). The(black/white) x denotes the (numerical/theoretical) center of thesource.


Figure 3.14. Back-propagation results for the geometry sqReg2with perturbed boundary (from left-to-right, ε = 0.01, 0.025 and0.05). The (black/white) x denotes the (numerical/theoretical)center of the source.

0.18 0.19 0.2 0.21 0.22−0.19

−0.18

−0.17

−0.16

−0.15

0.18 0.19 0.2 0.21 0.22−0.19

−0.18

−0.17

−0.16

−0.15

0.18 0.19 0.2 0.21 0.22−0.19

−0.18

−0.17

−0.16

−0.15

Figure 3.15. Arrival-time results for the geometry sqReg2 withperturbed boundary (from left-to-right, ε = 0.01, 0.025 and 0.05)


Algorithm Perturbation amplitude ε xest d(x0, xest)Kirchhoff 0.01 [0.2 -0.1625] 0.0125

0.025 [0.2 -0.1625] 0.01250.05 [0.1875 -0.15] 0.03

Back-propagation 0.01 [0.2125 -0.1625] 0.00790.025 [0.2 -0.1625] 0.01250.05 [0.1875 -0.15] 0.03

Arrival time 0.01 [0.2022 -0.1687] 0.00790.025 [0.1917 -0.167] 0.01860.05 [0.1944 -0.1643] 0.0166

Table 3.3. Numerical results for localization of the source at x0 =[0.21 −0.17] using sqReg2 geometry with boundary perturbations.

CHAPTER 4

Imaging in visco-elastic media obeying a frequency

power-law

Abstract. In this chapter we consider the problem of reconstructing a small

anomaly in a viscoelastic medium from wavefield measurements. We choose

Szabo’s model [95] to describe the viscoelastic properties of the medium.

Expressing the ideal elastic field without any viscous effect in terms of themeasured field in a viscous medium, we generalize the imaging procedures

in Chapter 2 to detect an anomaly in a visco-elastic medium from wavefieldmeasurements.

4.1. Introduction

In Chapter 2 we have considered anomaly imaging in a purely quasi-incompressibleelastic medium. In this chapter, we consider the problem of reconstructing a smallanomaly in a viscoelastic medium from wavefield measurements. The Voigt modelis a common model to describe the viscoelastic properties of tissues. Catheline etal. [42] have shown that this model is well adapted to describe the viscoelasticresponse of tissues to low-frequency excitations. We choose a more general modelderived by Szabo [95] that describes observed power-law behavior of many vis-coelastic materials. It is based on a time-domain statement of causality. It reducesto the Voigt model for the specific case of quadratic frequency loss. Expressingthe ideal elastic field without any viscous effect in terms of the measured field in aviscous medium, we generalize the methods described in Chapter 2 to recover theviscoelastic and geometric properties of an anomaly from wavefield measurements.

The chapter is organized as follows. In Section 4.2 we introduce a general visco-elastic wave equation. Section 4.3 is devoted to the derivation of the Green functionin a viscoelastic medium. In Section 4.4 we present anomaly imaging procedures invisco-elastic media.

4.2. General visco-elastic wave equation

When a wave travels through a biological medium, its amplitude decreases withtime due to attenuation. The attenuation coefficient for biological tissue may beapproximated by a power-law over a wide range of frequencies. Measured attenua-tion coefficients of soft tissue typically have linear or greater than linear dependenceon frequency [95].

In an ideal medium, i.e., without attenuation, Hooke’s law expresses the fol-lowing relationship between stress and strain tensors:

(4.1) T = C : S,65

66 4. IMAGING IN VISCO-ELASTIC MEDIA

where T , C and S are respectively the stress, the stiffness and the strain tensor oforders 2, 4 and 2 and : represents the tensorial product.

Consider a dissipative medium. Suppose that the medium is homogeneous andisotropic. We write

C = [Cijkl] = [λδijδkl + µ(δikδjl + δilδjk)] ,(4.2)

η = [ηijkl] = [ηsδijδkl + ηp(δikδjl + δilδjk)] ,(4.3)

where δ is the Kronecker delta function, µ, λ are the Lame parameters, and ηs, ηp arethe shear and bulk viscosities, respectively. Here we have adopted the generalizedsummation convention over the repeated index.

Throughout this chapter we suppose that

(4.4) ηp, ηs << 1.

For a medium obeying a power-law attenuation model and under the smallnesscondition (4.4), a generalized Hooke’s law reads [95]

(4.5) T (x, t) = C : S(x, t) + η : M(S)(x, t)

where the convolution operator M is given by

(4.6) M(S) =

−(−1)y/2 ∂y−1S∂ty−1 y is an even integer,

2π (y − 1)!(−1)(y+1)/2H(t)

ty ∗ S y is an odd integer,

− 2πΓ(y) sin(yπ/2)H(t)

|t|y ∗ S y is a non integer.

Here H(t) is the Heaviside function and Γ denotes the gamma function.Note that for the common case, y = 2, the generalized Hooke’s law (4.5) reduces

to the Voigt model,

(4.7) T = C : S + η :∂S∂t.

Taking the divergence of (4.5) we get

∇ · T =(λ+ µ

)∇(∇ · u) + µ∆u,

where

λ = λ+ ηpM(·) and µ = µ+ ηsM(·).Next, considering the equation of motion for the system, i.e.,

(4.8) ρ∂2u

∂t2− F = ∇ · T ,

with ρ being the constant density and F the applied force and using the expressionfor ∇ · T , we obtain the generalized visco-elastic wave equation

(4.9) ρ∂2u

∂t2− F =

(λ+ µ

)∇(∇ · u) + µ∆u.

4.3. Green’s function

In this section we find the Green function of the viscoelastic wave equation(4.9). For doing so, we first need a Helmholtz decomposition.

4.3. GREEN’S FUNCTION 67

4.3.1. Helmholtz decomposition. The following lemma holds.

Lemma 4.1. If the displacement field u(t, x) satisfies (4.9) and if the body forceF = ∇ϕf + ∇× ψf then there exist potentials ϕu and ψu such that

• u = ∇ϕu + ∇× ψu; ∇ · ψu = 0;

• ∂2ϕu

∂t2 =ϕf

ρ +c2p∆ϕu+νpM(∆ϕu) ≈ ϕf

ρ − νpM(ϕf )ρc2p

+c2p∆ϕu+νp

c2pM(∂2

t ϕu);

• ∂2ψu

∂t2 =ψf

ρ +c2s∆ψu+νsM(∆ψu) ≈ ψf

ρ − νsM(ψf )ρc2s

+c2s∆ψu+ νs

c2sM(∂2

t ψu),

with

c2p =λ+ 2µ

ρ, c2s =

µ

ρ, νp =

ηp + 2ηsρ

, and νs =ηsρ.

Let

(4.10) Km(ω) = ω

√(1 − νm

c2mM(ω)), m = s, p,

where the multiplication operator M(ω) is the Fourier transform of the convolutionoperator M.

Supposing that ϕu and ψu are causal implies the causality of the inverse Fouriertransforms of Km(ω),m = s, p. Applying the Kramers-Kronig relations, it followsthat(4.11)

−=mKm(ω) = H[<eKm(ω)

]and <eKm(ω) = H

[=mKm(ω)

], m = p, s,

where H is the Hilbert transform. Note that H2 = −I. The convolution operator Mgiven by (4.6) is based on the constraint that causality imposes on (4.5). Under thesmallness assumption (4.4), the expressions in (4.6) can be found from the Kramers-Kronig relations (4.11). One drawback of (4.11) is that the attenuation, =mKm(ω),must be known at all frequencies to determine the dispersion, <eKm(ω). However,bounds on the dispersion can be obtained from measurements of the attenuationover a finite frequency range [85].

4.3.2. Solution of (4.9) with a concentrated force. Let uij denote thei-th component of the solution uj of the elastic wave equation related to a force Fconcentrated in the xj-direction. Let j = 1 for simplicity and suppose that

(4.12) F = T (t)δ(x− ξ)e1 = T (t)δ(x− ξ)(1, 0, 0),

where ξ is the source point and (e1, e2, e3) is an orthonormal basis of R3. The

corresponding Helmholtz decomposition of the force F can be written as

(4.13)

F = ∇ϕf + ∇× ψf ,

ϕf = −T (t)4π

∂∂x1

(1r

),

ψf = T (t)4π

(0, ∂

∂x3

(1r

),− ∂

∂x2

(1r

)),

where r = |x− ξ| [88].Consider the Helmholtz decomposition for u1 as

(4.14) u1 = ∇ϕ1 + ∇× ψ1,


where ϕ1 and ψ1 are respectively the solutions of the equations

∆ϕ1 −1

c2p

∂2ϕ1

∂t2+νpc4p

M(∂2t ϕ1) =

ϕfc2pρ

− νpM(ϕf )

ρc4p,(4.15)

∆ψ1 −1

c2s

∂2ψ1

∂t2+νsc4s

M(∂2t ψ1) =

ψfc2sρ

− νsM(ψf )

ρc4s.(4.16)

Taking the Fourier transform of (4.14),(4.15) and (4.16) with respect to t weget

u1 = ∇ϕ1 + ∇× ψ1(4.17)

∆ϕ1 +K2p(ω)

c2pϕ1 =

ϕfρc2p

− νpM(ω)ϕfρc4p

,(4.18)

∆ψ1 +K2s (ω)

c2sψ1 =

ψfρc2s

− νsM(ω)ψfρc4s

,(4.19)

with Km(ω),m = p, s, given by (4.10).It is well known that the Green functions of the Helmholtz equations (4.18)

and (4.19) are

gm(r, ω) =e√−1

Km(ω)cm

r

4πr, m = s, p.

Therefore, following [88], we get the following expression for ϕ1:

(4.20) ϕ1(x, ω; ξ) = − 1

ρc2p(1 − νpM(ω)

c2p)T (ω)

4πρ

∂

∂x1

(1

r

)∫ r/cp

0

ζe√−1Kp(ω)ζ dζ.

In the same way, the vector ψ1 is given by(4.21)

ψ1(x, ω; ξ) =1

ρc2s(1−νsM(ω)

c2s)T (ω)

4πρ

(0,

∂

∂x3

(1

r

),− ∂

∂x2

(1

r

))∫ r/cs

0

ζe√−1Ks(ω)ζ dζ.

Introduce the following notation:

Im(x, ω) = Am

∫ r/cm

0

ζe√−1Km(ω)ζ dζ(4.22)

Em(x, ω) = Ame√−1Km(ω) r

cm ,(4.23)

Am(ω) = (1 − νmM(ω)

c2m), m = p, s.(4.24)

We obtain, after a lengthy but simple calculation, that ui1 is given by

ui1 = T (ω)4πρ

∂2

∂xix1

(1r

)[Is(r, ω) − Ip(r, ω)] + T (ω)

4πρc2pr∂r∂xi

∂r∂x1

Ep(r, ω)

+ T (ω)4πρc2sr

(δi1 − ∂r

∂xi

∂r∂x1

)Es(r, ω),

and therefore, it follows that the solution uij for an arbitrary j is

uij = T (ω)4πρ (3γiγj − δij)

1r3 [Is(r, ω) − Ip(r, ω)] + T (ω)

4πρc2pγiγj

1rEp(r, ω)

+ T (ω)4πρc2s

(δij − γiγj)1rEs(r, ω),

4.4. IMAGING PROCEDURE 69

where γi = (xi − ξi)/r.

4.3.3. Green’s function. If we substitute T (t) = δ(t), where delta is theDirac mass, then the function uij = Gij is the i-th component of the Green function

related to the force concentrated in the xj-direction. In this case we have T (ω) = 1.

Thus we have the following expression for Gij :

Gij = 14πρ (3γiγj − δij)

1r3 [Is(r, ω) − Ip(r, ω)] + 1

4πρc2pγiγj

1rEp(r, ω)

+ 14πρc2s

(δij − γiγj)1rEs(r, ω),

which implies that

(4.25) Gij(r, ω; ξ) = gpij(r, ω) + gsij(r, ω) + gpsij (r, ω),

where

(4.26) gpsij (r, ω) =1

4πρ(3γiγj − δij)

1

r3[Is(r, ω) − Ip(r, ω)] ,

(4.27) gpij(r, ω) =Ap(ω)

ρc2pγiγj g

p(r, ω),

and

(4.28) gsij(r, ω) =As(ω)

ρc2s(δij − γiγj) g

s(r, ω).

Let G(r, t; ξ) = (Gij(r, t; ξ)) denote the transient Green function of (4.9) asso-ciated with the source point ξ. Let Gm(r, t; ξ) and Wm(r, t) be the inverse Fouriertransforms of Am(ω)gm(r, ω) and Im(r, ω),m = p, s, respectively. Then, from (4.25-4.28), we have

Gij(r, t; ξ) = 1ρc2p

γiγjGp(r, t; ξ) + 1

ρc2s(δij − γiγj)G

s(r, t; ξ)

+ 14πρ (3γiγj − δij)

1r3 [Ws(r, t) −Wp(r, t)] .

Note that by a change of variables,

Wm(r, t) =4π

c2m

∫ r

0

ζ2Gm(ζ, t; ξ)dζ.

4.4. Imaging procedure

Consider the limiting case λ→ +∞. The Green function for a quasi-incompressiblevisco-elastic medium is given by

Gij(r, t; ξ) = 1ρc2s

(δij − γiγj)Gs(r, t; ξ)

+ 116π2ρc2s

(3γiγj − δij)1r3

∫ r0ζ2Gs(ζ, t; ξ)dζ.

To generalize the detection algorithms presented in Chapter 2 to the visco-elasticcase we shall express the ideal Green function without any viscous effect in termsof the Green function in a viscous medium. From

Gs(r, t; ξ) =1√2π

∫

R

e−√−1ωtAs(ω)gs(r, ω) dω,


it follows that

Gs(r, t; ξ) =1√2π

∫

R

As(ω)e√−1(−ωt+ Ks(ω)

csr)

4πrdω.

4.4.1. Approximation of the Green function. Introduce the operator

Lφ(t) =1

2π

∫

R

∫ +∞

0

As(ω)φ(τ)e√−1Ks(ω)τe−

√−1ωt dτ dω,

for a causal function φ. We have

Gs(r, t; ξ) = L(δ(τ − r/cs)

4πr),

and therefore,

L∗Gs(r, t; ξ) = L∗L(δ(τ − r/cs)

4πr),

where L∗ is the L2(0,+∞)-adjoint of L.

Consider for simplicity the Voigt model. Then, M(ω) = −√−1ω and hence,

Ks(ω) = ω

√

1 +

√−1νsc2s

ω ≈ ω +

√−1νs2c2s

ω2,

under the smallness assumption (4.4). The operator L can then be approximatedby

Lφ(t) =1

2π

∫

R

∫ +∞

0

As(ω)φ(τ)e− νs

2c2sω2τ

e√−1ω(τ−t) dτ dω.

Since

∫

R

e− νs

2c2sω2τ

e√−1ω(τ−t) dω =

√2πcs√νsτ

e−c2s(τ−t)2

2νsτ ,

and√−1

∫

R

ωe− νs

2c2sω2τ

e√−1ω(τ−t) dω = −

√2πcs√νsτ

∂

∂te−

c2s(τ−t)2

2νsτ ,

it follows that

(4.29) Lφ(t) =

∫ +∞

0

t

τφ(τ)

cs√2πνsτ

e−c2s(τ−t)2

2νsτ dτ.

Analogously,

(4.30) L∗φ(t) =

∫ +∞

0

τ

tφ(τ)

cs√2πνst

e−c2s(τ−t)2

2νst dτ.

Since the phase in (4.30) is quadratic and νs is small then by the stationaryphase theorem 4.2, we can prove that

L∗φ ≈ φ+νs2c2s

∂tt(tφ), Lφ ≈ φ+νs2c2s

t∂ttφ,

and

(4.31) L∗Lφ ≈ φ+νsc2s∂t(t∂tφ),


and therefore,

(4.32) (L∗L)−1φ ≈ φ− νsc2s∂t(t∂tφ).

4.4.2. Reconstruction methods. From the previous section, it follows thatthe ideal Green function, δ(τ − r/cs)/(4πr), can be approximately reconstructedfrom the viscous Green function, Gs(r, t; ξ), by either solving the ODE

φ+νsc2s∂t(t∂tφ) = L∗Gs(r, t; ξ),

with φ = 0, t 0 or just making the approximation

δ(τ − r/cs)/(4πr) ≈ L∗Gs(r, t; ξ) − νsc2s∂t(t∂tL

∗Gs(r, t; ξ)).

Once the ideal Green function is reconstructed, one can find its source ξ usingthe algorithms in Chapter 2. One can also find the shear modulus of the anomalyusing the ideal near-field measurements which can be reconstructed from the near-field measurements in the viscous medium.


For the following illustrations, we take ρ = 1000, cs = 1, cp = 40, r = 0.015and νp = 0.

Figure 1 : We plot, for differents values of y and νsthe function

t→ 1

ρc2p(Gp(r, t; ξ) +Gs(r, t; ξ)) +

1

4πρr3[Ws(r, t) −Wp(r, t)] .

Figure 2 : We plot, for differents values of y and νs at t = 0.015 the function

(x, y) → 1

ρc2p

((x/r)2Gp(r, t; ξ) + (1 − (x/r)2)Gs(r, t; ξ)

)+

1

4πρr3(3(x/r)2−1) [Ws(r, t) −Wp(r, t)] .

Figure 3 : For φ(t) = exp(−50 ∗ (t − 1).2)′′, an L∞-error between Lφ andφ+ νs

2c2stφ′′ is ploted : we observe an error of two, as expected by stationary phase

theorem.


In this chapter we have computed the Green function in a visco-elastic mediumobeying a frequency power-law. For the Voigt model, which corresponds to a qua-dratic frequency loss, we have used the stationary phase theorem to reconstructthe ideal Green function from the visco-elastic one by solving an ODE. Once theideal Green function is reconstructed, one can find its source ξ using the algorithmsin Chapter 2. For more general power-law media, one can recover the ideal Greenfunction from the visco-elastic one by inverting a fractional derivative operator.


0 0.005 0.01 0.015 0.02 0.025 0.03−0.5

0

0.5

1

1.5

2

0 0.005 0.01 0.015 0.02 0.025 0.03−0.5

0

0.5

1

1.5

2

0 0.005 0.01 0.015 0.02 0.025 0.03−0.5

0

0.5

1

1.5

2

Figure 4.1. Temporal response to a spatiotemporal delta function using a

purely elastic Green’s function (red line) and a viscous Green’s function (blueline) : Left, y = 1.5, νs = 4 ; Center, y = 2, νs = 0.2 ; Right, y = 2.5,

νs = 0.002. 2

Appendix A: Proof of the approximation formula

The proof of formula (4.31) is based on the following theorem (see [65, Theorem7.7.1]).

Theorem 4.2. (Stationary Phase)Let K ⊂ [0,∞) be a compact set, X anopen neighborhoud of K and k a positive integer. If ψ ∈ C2k

0 (K), f ∈ C3k+1(X)and Im(f) ≥ 0 in X, Im(f(t0)) = 0, f ′(t0) = 0, f ′′(t0) 6= 0, f ′ 6= 0 in K \ t0then for ε > 0∣∣∣∣∣∣

∫

K

ψ(t)eif(t)/εdx− eif(t0)/ε (λf ′′(t0)/2πi)−1/2

∑

j<k

εjLjψ

∣∣∣∣∣∣≤ Cεk

∑

α≤2k

sup |ψ(α)(x)|.

Here C is bounded when f stays in a bounded set in C3k+1(X) and |t− t0|/|f ′(t)|has a uniform bound. With,

gt0(t) = f(t) − f(t0) −1

2f ′′(t0)(t− t0)

2,

which vanishes up to third order at t0, we have

Ljψ =∑

ν−µ=j

∑

2ν≥3µ

i−j2−ν

ν!µ!(−1)νf ′′(t0)

−ν(gµt0ψ)(2ν)(t0). 2

APPENDIX A: PROOF OF THE APPROXIMATION FORMULA 73

Figure 4.2. 2D spatial response to a spatiotemporal delta function at t =

0.015 with a purely elastic Green’s function, a viscous Green’s function withy = 2, νs = 0.2 and y = 2.5, νs = 0.002. 2

0 0.2 0.4 0.6 0.8 1−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

t

νs / c

s2 = 0.001

φL φφ + ν

s/(2c

s2) t φ’’

−11 −10 −9 −8 −7 −6 −5−11

−10

−9

−8

−7

−6

−5

−4

−3

−2

log(νs / c

s2)

|Err

eur

| ∞

Figure 4.3. Approximation of L via stationary phase theorem : Left, com-parison between Lφ and φ+ νs

c2stφ′′ where νs

c2s

= 0.0001 and φ is a mexican hat,

Right, error νsc2s→ ‖Lφ − φ + νs

c2s‖∞ in logarithmic scale. 2

Note that L1 can be expressed as the sum L1ψ = L11ψ+L2

1ψ+L31ψ, where Lj1

is respectively associate to the pair (νj , µj) = (1, 0), (2, 1), (3, 2) and is identified to

L11ψ = −1

2i f′′(t0)−1ψ(2)(t0),

L21ψ = 1

222!if′′(t0)−2(gt0u)

(4)(t0) = 18if

′′(t0)−2(g(4)t0 (t0)ψ(t0) + 4g

(3)t0 (t0)ψ

′(t0)),

L31ψ = −1

232!3!if′′(t0)−3(g2

t0ψ)(6)(t0) = −1232!3!if

′′(t0)−3(g2t0)

(6)(t0)ψ(t0).


Now we turn to the proof of formula (4.31). Let us first consider the case ofoperator L∗. We have

L∗φ(t) =

∫ +∞

0

τ

tφ(τ)

cs√2πνst

e−c2s(τ−t)2

2νst dτ =1

t√ε

(∫ +∞

0

ψ(τ)eif(τ)/ε

),

with, f(τ) = iπ(τ − t)2, ε = 2πνstc2s

and ψ(τ) = τφ(τ). Remark that the phase f

satisfies at τ = t , f(t) = 0, f ′(t) = 0, f ′′(t) = 2iπ 6= 0. Moreover, we have

eif(t)/ε(ε−1f ′′(t)/2iπ

)−1/2=

√ε

gt(τ) = f(τ) − f(t) − 12f

′′(t)(τ − t)2 = 0

L1ψ(t) = L11ψ(t) = −1

2i f′′(t)−1ψ

′′(t) = 1

4π (tφ)′′.

Thus, Theorem 4.2 implies that∣∣∣∣L∗φ(t) −

(φ(t) +

νs2c2s

(tφ)′′)∣∣∣∣ ≤

C

tε3/2

∑

α≤4

sup |(tφ)(α)|.

The case of the operator L is very similar. Note that

Lφ(t) =

∫ +∞

0

t

τφ(τ)

cs√2πνsτ

e−c2s(τ−t)2

2νsτ dτ =t√ε

(∫ +∞

0

ψ(τ)eif(τ)/ε

),

with f(τ) = iπ (τ−t)2τ , ε = νs

2πc2sand ψ(τ) = φ(τ)τ−

32 . It follows that

f ′(τ) = iπ

(1 − t2

τ2

), f ′′(τ) = 2iπ

t2

τ3, f ′′(t) = 2iπ

1

t,

and the function gt(τ) equals to

gt(τ) = iπ(τ − t)2

τ− iπ

(τ − t)2

t= iπ

(t− τ)3

τt.

We deduce that(gtψ)(4)(t) =

(g(4)t (t)ψ(t) + 4g

(3)t (t)ψ′(t)

)= iπ

(24t3 ψ(t) − 24

t2 ψ′(t))

(g2tψ)(6)(t) = (g2

t )(6)(t)ψ(t) = −π2 6!

t4ψ(t),

and then,

L11ψ = −1

i

(12 (f ′′(t))−1ψ′′(t)

)= 1

4π t(φ√t

)′′= 1

4π

(√tφ′′(t) − φ′(t)√

t+ 3

4φt3/2

)

L21ψ = 1

8if′′(t)−2

(g(4)t (s)ψ(s) + 4g

(3)t (t)ψ′(t)

)= 1

4π

(3(φ(t)√t

)′− 3 φ(t)

t3/2

)= 1

4π

(3 φ

′(t)√t

− 92φ(t)t3/2

)

L31ψ = −1

232!3!if′′(t)−3(g2

t )(6)(t)ψ(s) = 1

4π

(154φ(t)t3/2

),

where φ(τ) = φ(τ)/τ . Then, we have

L1ψ = L11ψ + L2

1ψ + L31ψ

=1

4π

(√tφ′′(t) + (3 − 1)

φ′(t)√t

+

(3

4− 9

2+

15

4

)φ(t)

t3/2

)=

1

4π√t

(tφ(t)

)′′=

1

4π√tφ′′(t),

and again Theorem 4.2 shows that∣∣∣∣Lφ(t) −(φ(t) +

νs2c2s

tφ′′(t)

)∣∣∣∣ ≤ Ctε3/2∑

α≤4

sup |ψ(α)(t)|.

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Index

arrival time, 43, 46, 47, 53

asymptotic expansions, 9, 31

attenuation, 65, 67

back-propagation, 20, 43, 45, 53

dispersion, 67

far-field measurements, 7, 15–17, 27, 29,

39–41

focal spot, 19, 27, 41

frequency power-law, 65, 66

geometrical control method, 43, 44, 51

Green function, 28, 41, 65, 66

Hashin-Shtrikman bounds, 14

Helmholtz decomposition, 66

Helmholtz equation, 8, 68

Hilbert Uniqueness Method, 50

inner expansion, 12, 15, 35, 39

Kirchhoff algorithm, 20, 43, 45, 49, 53

Kramers-Kronig relations, 67

layer-potentials, 8, 29

limited-view data, 43

magneto-acoustic current imaging, 48

modified Stokes system, 27, 29

MUSIC algorithm, 43, 46, 53

near-field measurements, 7, 15, 17, 27, 39

outer expansion, 16, 19, 40

photo-acoustic imaging, 48

polarization tensor, 14, 20

quasi-incompressible elasticity, 27

radiation force imaging, 7, 47

resolution limit, 17, 19, 41–43

signal-to-noise ration, 21

time-reversal, 8, 17, 27, 41

visco-elastic wave equation, 65

viscous moment tensor, 36

Voigt model, 42, 65, 66, 70

wave equation, 8, 15, 43, 44

81

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